Controlling volatility of wind-solar power

Apart from nuclear power and hydropower (power from biomass and waste could be mentioned, too), the conventional electric power production by gas and fossil fuel power stations generates $CO_{2}$ as a byproduct. Nuclear power plants do not have this problem, but they have other disadvantages, in particular production of radioactive waste. All these problems do not appear when electric power is produced by solar panels and wind turbines alone. Nevertheless, wind-solar power has serious disadvantages too, the most serious one being the volatile energy production: Weather conditions change rapidly and as a consequence energy production of wind-solar power fluctuates considerably. How serious the consequences are, depends on two factors: i) the strength of the volatility, ii) the volatility that a consumer can tolerate - cf. the key phrase “new thinking”[4].

The problem of volatility is always a consequence of the mismatch between electric power production and electric power consumption. If we just consider passive storage devices, like pumped-storage plants, for example, the strength of volatility can be determined in the following way. We divide the wind-solar energy production $P_{v}$ and the energy demand $P_{d}$ into two parts: the average parts $P_{va}$ and $P_{da}$ being constant over the year and the fluctuating parts $P_{vf}$ and $P_{df}$. The power $P_{sf}$ streaming into and out of storage devices has a fluctuation part only since the devices are passive. The average of each fluctuating part is zero. Therefore we get the simple equations

$$P_{va}=P_{da}$$

(1)

and therefore,

$$\begin{array}[]{ccc}P_{sf}&=&P_{vf}-P_{df}\end{array}$$

(2)

Integrating these powers yields the energies

$$\begin{array}[]{ccc}E_{v(d)}(t)&=&E_{va(da)}(t)+E_{vf(df)}(t)\\ E_{va(da)}(t)&=&P_{va(da)}\cdot t\\ E_{vf(df)}(t)&=&\intop_{0}^{t}P_{vf(df)}dt\\ E_{sf}&=&E_{vf}-E_{df}\end{array}$$

(3)

The required storage $E_{sfmax}$ is obtained from the expression

$$E_{sfmax}=\max_{t}\{E_{sf}(t)\}-\min\{E_{sf}(t)\}$$

(4)

In order to obtain these functions for 2019, we have used the data of ref[3] which contain power measurements every $15$ minutes [in MW] for electric load and the volatile power, consisting of: offshore and onshore windpower as well as solar power. The result is obtained directly for the demand (load): $P_{da}=56.4$GW and the fluctuation part $E_{df}$ is plotted in Fig.1. The correct values for $P_{va}$ and $P_{vf}$ require a moment of consideration. The measurement data lead to the value $\hat{P}_{va}=18.9$GW.

Since we intend to satisfy the average electric energy demand by wind-solar power alone, we need an average electric power production of at least $P_{va}=P_{da}=\text{56.4}$GW as well. To manage this problem, we assume that the distribution of solar cells and wind turbines is already at its optimum in Germany. In that case we can easily estimate the situation where all average electric energy is delivered by wind-solar power: We just have to multiply the average wind-solar power and its fluctuation by a scaling factor[1]. Clearly the distribution of wind-solar power is not at its optimum in reality. Therefore, the scaling is an approximation. However, understanding the trends is the aim of this paper and for that purpose the scaling will be a sufficiently good approximation. As the distribution becomes increasingly optimized, such an approximation will steadily improve. Now taking as natural scaling factor $56.4/18.9$ for wind-solar power we get

$$\begin{array}[]{ccc}P_{va}&=&\hat{P}_{va}\cdot\frac{56.4}{18.9}=P_{da}\\ \\ P_{vf}&=&\hat{P}_{vf}\cdot\frac{56.4}{18.9}\end{array}$$

(5)

where $\hat{P}_{va}$ and $\hat{P}_{vf}$ are obtained from the measurement data. In the same way we get the scaled integrated quantities $E_{v},$$E_{va}$ and $E_{vf}$ after scaling the original mathematical expressions with the same factor. Together with the scaling factor $56.4/18.9$ those expressions will be used throughout this paper. Of course, wind-solar and demand data differ from year to year, but, considered over the year 2019, the typical properties should be similar enough to reveal the relevant trends. For a detailed technical analysis, the observation period of one year does, of course, not suffice. Instead, observation periods over many years are necessary. This is possible, but beyond the scope of the present paper.

As a first application we calculate the storage capacity necessary to fulfill eq.[2] and we find: the required electric storage removing volatility is $75.9TWh$, cf. eq.4 and Fig.1.

This result proves that volatility is a serious problem. Indeed volatility led Sinn to the conclusion that energy production resting essentially on wind-solar power alone will take us into an economic nirvana[5]. One could argue that even a total storage capacity of about $84$TWh[1], [2] is in principle feasible by transforming the huge Norwegian hydro dams into pumped-storage plants. However, two facts are obvious: i) The present electric power production has to be multiplied by a factor of about $5$ [1], if all transportation, warm water, space heating and a considerable percentage of process heating are switched to electric power as well. With the configuration presented here so far, this is impossible. ii) Even if we do not consider transportation, warm water, space heating, and process heating, the pump storage capacity requirements would be so enormous that an export of this wind-solar scheme to many other nations would be out of the question - a bitter disadvantage when Germany wants to be a forerunner.

How can we successfully proceed? Of course we need a kind of active buffering. Sinn [1] suggests various combinations between wind-solar power and conventional power plants. This works for the present needs but eliminates the advantages of using wind-solar power alone. Furthermore, this scheme becomes questionable if the needs of electric power are five times greater than now. And that will happen if energy production of Germany will be switched to electric power generation as far as possible.

In this paper we discard such schemes. Instead we suggest a combination of three methods: First, creating a surplus of wind-solar power. Second, applying smart meters[4]. Both methods are described in section II. We show that through these methods the storage capacity requirements are reduced by a factor of about 50 or even become marginal. Third, applying new criteria for optimizing the efficiency of wind turbines, solar cells and their distribution across the country. We show in section III that through these additional methods the smart meters need distinctively less flexibility. We think that these results give us the justification for extrapolating to the case where in addition to the present electrical energy production all energy for the total transport, warm water, space heating and and a considerable percentage of process heating is exclusively obtained from wind-solar power. This is discussed in section IV. Our conclusions are presented at the end of the paper.

In this section we discuss the situation, in which the generation of electric power is taken over by wind-solar power alone.

In contrast to simple passive buffers like pumped-storage plants with all their capacity limitations active buffers like power plants can satisfy the demand necessary for guaranteeing a safe power delivery. To avoid $CO_{2}$ production we choose as active buffers wind-solar power itself. Assuming[1] as above an already optimum distribution of wind-solar power devices across the nation, the surplus of wind-solar power can be expressed again by a scaling factor , the strength $\alpha$, and we get for the wind-solar energy

$$E_{v}(t)\longrightarrow(1+\alpha)E_{v}(t),\alpha=const$$

The price to be paid for this scheme is a reduced efficiency. This is all the more the case since at times of low wind-solar power the surplus of wind-solar power is reduced as well enforcing a larger $\alpha$ value than expected from the average gain in power. To keep $\alpha$ within reasonable limits we need the concept[6] and on a large scale the application[7] of smart meters[4]. Such devices control the electrical consumption very effectively by setting higher consumption prices if less power is available and lower prices if there is a surplus of power. Thus smart meters act like passive buffering devices by moving the peaks of electric consumption to the peaks of wind-solar power. We note in passing that moving electric consumption on an hourly basis or less has the same effect as smoothing, cf. ref. [1], section 4.

Of course a detailed simulation of smart meters is intricate. However, we think that the following simulation of smart meters reflects the principle effects satisfactorily: $E_{d}(t)$ is the energy of electric consumption as function of time. Now, if wind-solar production is not sufficient, it produces energy corresponding to a demand $E_{d}(t^{\prime})<E_{d}(t)$. The smart meters have now the task, by increasing prices for $1kWh$ to enforce a reduced demand, namely $E_{d}(t^{\prime})$. Clearly that is always possible - due to exorbitant prices, if necessary. But to avoid such uneconomic incidents we introduce the delay time $\tau\geqslant 0$ and require $t^{\prime}>t-\tau$. The analogous happens for a surplus of wind-solar power: If there is an excess production of electric energy, corresponding to $E_{d}(t^{\prime})>E_{d}(t)$, smart meters charge low prices and $E_{d}$ is again not that of time t but that of $t^{\prime}$ and $t^{\prime}<t+\tau$. This means the strict requirement $t^{\prime}=t$ is replaced by

$$E_{d}(t-\tau)<E_{d}(t^{\prime})<E_{d}(t+\tau),\,\mid t^{\prime}-t\mid\leq\tau$$

(6)

As is the case in the applications of real smart meters, the electric power consumption becomes more flexible in this simulation. The produced energy till time t need not be $E_{d}(t)$ exactly but staying in the interval of eq.(6) suffices. This flexibility saves storage capacity as can be seen from an extreme (hypothetical) case: If $\tau$ becomes sufficiently big the required storage capacity approaches zero. Of course such large values of $\tau$ are unrealistic. However, values of hours or even days can be acceptable. We assume that a limit $\tau\leq 1$day is a very reasonable one. And note, this simulation of smart meters fulfills the criterion that - as in real smart meters - consumption of energy is only moved, no energy is generated or lost.

To avoid large $\tau$ values, finite $\alpha$-values and possibly electric buffering devices are still needed. We find out the relation between delay time $\tau$, capacity $E_{sfmax}$, and strength $\alpha$ in the following manner:

We fix $\alpha$ and $\tau$, proceed in time steps $\Delta t=15$min and define $E_{d}(t^{\prime})=\breve{E_{d}}$. At the beginning we set $\breve{E}=0$ and $E_{sf}=0$. Now at time $t+\Delta t$ we get

$$\begin{array}[]{ccc}\delta E_{v}&=&(1+\alpha)P_{v}(t)\cdot\Delta t\end{array}$$

Now three cases have to be distinguished (we set $s=t+\Delta t$):

i) $E_{d}(s-\tau)\leq\breve{E}_{d}+\delta E_{v}\leq E_{d}(s+\tau)$, then $\breve{E}_{d}\rightarrow\breve{E}_{d}+\delta E_{v}$.

ii) $\breve{E}_{d}+\delta E_{v}>E_{d}(s+\tau)$, then $\breve{E}_{d}\rightarrow E_{d}(s+\tau)$ and $E_{sf}\rightarrow E_{sf}+\breve{E}_{d}+\delta E_{v}-E_{d}(s+\tau)$

iii) $\breve{E}_{d}+\delta E_{v}<E_{d}(s-\tau)$, then $\breve{E}_{d}\rightarrow E_{d}(s-\tau)$ and $E_{sf}\rightarrow E_{sf}+\breve{E}_{d}+\delta E_{v}-E_{d}(s-\tau)$

Furthermore, to save storage, we enforce $E_{sf}\leq 0$ by replacing a positive $E_{sf}$ with zero. Thus a positive $E_{sf}$ becomes a kind of ’wasted’ energy that has to be removed somehow (see below). At the end of the calculation $E_{sfmax}$ is given by

$$E_{sfmax}=\max_{t}\{-E_{sf}(t)\}$$

Repeating this procedure for various $\tau$ values we get the function
$E_{sfmax}(\tau,\alpha)$ and from that function the inverse function $\tau(E_{sfmax},\alpha)$.

Results of our calculations are shown in Fig.2. We have plotted $\tau(E_{sfmax},\alpha)$ for various fixed $\alpha$ values. The results contain the important conclusion that storage capacity can be replaced by a surplus of wind-solar power. This phenomenon opens a wide field of possibilities: If storage capacity does not represent a problem $\alpha=0.3$ might be sufficient. On the other hand storage capacity becomes rather uncritical for $\alpha\geqq 0.5$. And for $\alpha=1$ the storage capacity is no longer a problem.

The arising ’wasted’ energy need not be small at all. In fact, if all possible energy is generated the average power amounts to $(1+\alpha)\cdot 56.4GW$ and thus the average ’wasted’ power to $\alpha\cdot 56.4GW.$ To get rid of it directly is one possibility. This can easily be achieved by reducing the wind-solar power, as soon as ’wasted’ energy begins to build up. The advantage of this procedure would be a strain imposed on the electricity network that would not essentially be higher than for $\alpha=0$.

An alternative would be exploiting this ’wasted’ power for processes, e.g. for electrolytic and chemical processes. However one has to keep in mind that the surplus power is really extremely volatile as can be seen from Fig.3 for $\alpha=0.5$. Apart from high peaks there are - more important - periods, even weeks, where no ’wasted’ power is available.

The absolute costs per $kWh$ depend on assumptions, of how prices will develop in the future and which indirect costs have to be included in the calculation and which not. In fact, the estimates fluctuate strongly[1][8][9]. However, the relative increase of running costs per $kWh$ due to the ’wasted’ power can be assessed: For a small contribution of wind-solar power - so small, that peaks do not overshoot consumption - suppose running costs in the average to be $ws_{small}]$[€/kWh]. However, we do not deal with a small contribution. Instead an average demand of $56.4GW$ has to be satisfied. Applying the surplus power approach fulfilling this demand requires an average production of $(1+\alpha)\cdot 56.4GW$. Therefore, the increase of the running costs is $ws_{small}\rightarrow ws_{small}\cdot(1+\alpha)$ and the relative increase is given by $\alpha$.

At first sight it may seem obvious that wind-solar power should have its nominal power at high winds, at high sun radiation and moreover in regions with high winds and high sun-radiation, respectively. But the surplus wind-solar power becomes important if the wind-solar power production is weak and therefore weak-wind turbines and solar cells with good low-light performance will be essential for good additional power production. Weak-wind turbines having blades enlarged by a factor[10] $\sqrt{\beta}$, greater height[11] and thus higher wind speed enlarged by a factor[12] $\left(\gamma\right)^{1/3}$, provide an increase of power generation by a factor of $\beta\cdot\gamma$. We choose $\beta\cdot\gamma=2.$ This doubles the power production in the low-wind regime, $P_{l}=2\cdot P$. In the high wind regime, however, the power production saturates since these turbines have a reduced nominal power[12] $P_{nom}$. This justifies the ansatz

$$P_{l}(t)=P_{nom}\cdot\tanh\left(\beta\cdot\gamma\cdot P(t)/P_{nom}\right),\,\beta\cdot\gamma=2$$

Weak-light performance of solar cells[13] depends on the material used[14]. Mono-crystalline PV modules[15], multi junction[16] with selected band gaps and in the future the new generations of DSSCs[17][18] may have good weak light performance. And we assume that with good weak light performance the generated power can increase - as for the wind power - by a factor of $2$ also in the weak-light regime. (This approximation may be crude but less important as well, since the capacity factor of solar cells is dismal, at least in Germany[19], $10-13$ %, and therefore the dominant power generation will be that of wind turbines). So we get for the total surplus power (here defined as $P_{low}$) the ansatz

$$P_{low}(t)=P_{va}\cdot\tanh\left(2P_{v}(t)/P_{va}\right)$$

(7)

and

$$\begin{array}[]{ccc}\delta E_{v}&=&\left(P_{v}(t)+\alpha P_{low}(t)\right)\cdot\Delta t\end{array}$$

To demonstrate the importance of low energy production, we have selected a low nominal power $P_{nom}$ for $P_{low}$: $P_{nom}=P_{va}.$ And the average wind-solar power is about five times less than the normal nominal power.

The calculations correspond to those of the previous section. Results are presented in Fig. 4. The distinctly better outcome for the delay times $\tau$ is obvious in spite of the low nominal power $P_{va}$ of $P_{low}$. This proves the importance of good performance in weak wind and low light situations.

We found a similar improvement of the results when using offshore wind turbines. We repeated the calculations, only replacing the weak wind power by scaled offshore wind power. With a value of about $20$ the scaling factor guaranteeing $P_{va_{Off}}=P_{da}$ is rather large since at present the offshore power amounts to $5$% of the electric load (average) only, cf. Fig.4..

In the two preceding sections the possibility of applying solar-wind power without excessive use of storage devices has been demonstrated. However, we had only discussed the case of replacing the present electric energy production by wind-solar power. But this amounts to about $20\%\thickapprox 60GW$[1] over the year, whereas the total energy production amounts to $\thickapprox 280GW$, averaged over the year). $80\%$ consists of energy production for transport on a fossile basis, warm water, space heating and process heating. Converting this energy production into electric power should be possible not completely but to a large extent.

Therefore, the question is inescapable: Can all this electric power be generated by wind-solar power alone. I think a precise answer to this question presumes inspecting all the fluctuation data of wind and radiation across the various parts of Germany over many years. This is possible but beyond the scope of the present paper.

On the other hand, when we look at the required electric storage results for the various examples in the introduction we conclude that compared to the less drastic volatility of electric consumption the volatility of wind-solar power is the dominant part, cf. red dotted curve in Fig.1. But this part can be estimated by simple scaling as in section II, leading to a scale factor of $5$. This would mean that the plots in Fig.2 and Fig. 4 remain the same, only the storage values on the horizontal axis had to be multiplied by a factor of $5$. Since the storage values are not particularly critical, the effect of scaling would require a shift $\alpha\rightarrow.6$ or $\alpha\rightarrow 1$, which in our view seems to be a tolerable change.

Furthermore we can argue that in spite of its enormous volatility at least part of the ’wasted’ power can be used for chemical, in particular electrolytic processes, by which artificial fuel can be produced, e.g. for airplanes. This again would reduce the required wind-solar energy and thus the scaling factor.

It should also be pointed out that in any case the number of required wind turbines is enormous. After conversion into electric power an average power of $\thickapprox 250GW[1]$[1] has to be generated in Germany. Assuming i) a capacity factor of $25$% for wind turbines[18]) and ii) a $1/3$ contribution of solar power (probably a bit less[20]), a simple calculation leads to a required nominal wind power of $650000MW$. This means $430000$ {110000} wind turbines of the $1.5MW$ {6MW} type (height 120m {200m}) are needed to produce the average power. But this is not enough. In our approach this number has to be increased by $60$-100% to control volatility.

In view of these large numbers manufacture of wind turbines in mass production should be possible, reducing the cost of wind-solar power.

Is it possible to switch all present electric energy production of Germany to wind-solar power? This question has been answered in the negative by Sinn[5]. The main topic of this paper is to show that Sinn’s judgement is too pessimistic. When choosing a different ansatz, the results become different: We suggest marginalizing the strong volatility of wind-solar power by i) adding a substantial surplus of wind-solar power ii) installing smart meters[4],[6] iii) selecting different kinds of wind turbines, solar devices and switching to a good deal to offshore wind turbines.

The results of our ansatz are encouraging: The electric storage needed is reduced by more than $90\%$, even nearly $100$% should be possible.

The prize to be paid would be a $50$% - $100$% surplus of wind-solar power devices compared to the situation in which only the averaged wind-solar power production matches the averaged power consumption.

Our precise data[3] extend over the year 2019, a period sufficient to show that wind-solar power can be promising. Unambiguous results will be confirmed when weather conditions in Germany are carefully analyzed over a period of many years - but this is beyond the scope of the present paper. However, based on the present data, measured every 15 minutes in 2019, our approach avoiding excessive passive storage leads to the following conclusions: First, this approach is applicable to electric energy production in Germany and in other nations also, having no access to huge storage devices. Second, this approach leads to the prediction, that most of the present power demand of Germany could be supplied by wind-solar power alone. Third, this approach does no longer exclude the hope that even if most of the energy production in Germany is switching to electric energy - which means a factor of about $5$[1] -, this energy can be delivered by wind-solar power. In this case, however, no matter how we slice it, alone the number of required wind turbines would become tremendous: $430000$ [110000] wind turbines of the $1.5MW$[6MW] type (height 120m [200m]). Controlling the volatility according to our approach would increase these numbers further by $50-100$% . The running costs would rise by a factor $(1+\alpha),$ $\alpha$ being the strength of the additional electric power production, defined in section 2. According to our calculations reasonable values for $\alpha$ are in the range $[0.5,1.0]$.