Controlling Volatility of Wind-Solar Power In Germany

The main advantage of wind-solar power is the electric power production free of CO2. Its main disadvantage is the huge volatility of the system [national electric energy consumption powered by wind-solar power]. In fact, if this power production, averaged over one year, corresponds to the averaged electric consumption and is intended to replace all other electric power generating devices, then controlling the volatility of this system by using storage alone requires huge capacities of about 30TWh, capacities not available in Germany. However, based on German power data over the last six years (2015 till 2020) we show that the required storage capacity is decisively reduced, provided i) a surplus of wind-solar power is supplied, ii) smart meters are installed, iii) a different kind of wind turbines and solar panels is partially used, iv) a novel function describing this volatile system, is introduced. The new function, in turn, depends on three characteristic numbers, which means, that the volatility of this system is characterized by those numbers. When applying our schemes the results suggest that all the present electric energy in Germany can be obtained from controlled wind-solar power. And our results indicate that controlled wind-solar power can produce the energy for transportation, warm water, space heating and in part for process heating, requirering an increase of the electric energy production by a factor of 5. Then, however, a huge number of wind turbines and solar panels is required changing the appearance of German landscapes funamentally.

To smoothen the power flow of P v we apply a storage flow P sv = −P vf and with E sv (t) =ˆt P sv (t ′ )dt ′ the storage capacity needed is then Replacing v by d we get the condition for a smoothened load Putting consumption and volatile generation together, we get and Moreover, taking into account Eq.1, and Eq.2 we obtain after integration 139  140  141  142  143  144  145  146  147  148  149  150  151  152  153  154  155  156  157  158  159  160  161  162  163  164  165  166  167  168  169  170  171  172  173  174  175  176  177  178  179  180  181  182  183  184 Springer Nature 2021 L A T E X template 4 Controlling volatility of wind-solar power in Germany E s removes the mismatch between volatile consumption (E d ) and volatile energy (E v ), generated by wind-solar power.
In order to obtain the functions in the above equations, data of P v and P d are required. These are obtained from ref [5]. The data include those of the total electric load and the volatile electric power, consisting of: solar power as well as offshore and onshore windpower. Results are shown in Table 1, and we see two problems emerging. First, comparing the averaged total electric load of Germany, listed in column 2, with the corresponding electric volatile power, listed in column 3, it becomes obvious that Eq. [1] is violated. But since we intend to satisfy the electric energy demand by wind-solar power alone, we need the validity of this equation. Second, whereas the load does not change very much during the six years, the volatile wind-solar power increases by 50% and its various components change considerably. In particular the offshore wind power has increased by a factor of 3 during the six years. To avoid any difficulties connected with these sizeables changes, we treat each year separately and compare the results.
To manage the first 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 part by a scaling factor [2]. Undeniably the distribution of wind-solar power is in reality not at its optimum. Therefore, the scaling is an approximation. However, with increasing volatile power this approximation becomes better and better. And note, the contribution of volatile power has meanwhile passed the 35% mark.
With this in mind we calculate the above functions. Denoting the original volatile power, obtained from the measurement data, with P ν we get In the same way we get the scaled quantities E v ,E va and E vf . Note that the scaling factor is different for each year. For these scaled quantities the storage capacities smoothing the power flow, have been calculated. The results (in TWh) are in column 3 of Table 1 on the right side.
In a different scenario all power is produced e.g. by solar panels or by offshore or by onshore turbines alone. The scaling is quite analogous to the previous case and the required storage for smoothing the power can be found in columns 5 to 7.
The load has some volatility as well. The storage capacities suppressing fluctuations are depicted in column 2.
A different situation underlies the storage results in column 4. The volatile (scaled) wind-solar power drives the load that shows volatility too. The numbers required when using storage for faultless power transfer are found in this column. Note that this storage is of the same order of magnitude as the storage capacity required to smooth the (scaled) solar-wind power (column 3).   231  232  233  234  235  236  237  238  239  240  241  242  243  244  245  246  247  248  249  250  251  252  253  254  255  256  257  258  259  260  261  262  263  264  265  266  267  268  269  270  271  272  273  274  275  276 Springer Nature 2021 L A T E X template Indeed, volatility has led to the conclusion that energy production, resting essentially on wind-solar power alone, will take us into an economic nirvana [6].
It could be argued that even a total storage capacity of about 85 TWh [4] 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 [2]  to the case, where in addition to the present electric energy production all energy for the total transport, warm water, space heating and and a considerable percentage of process heating is exclusively obtained from windsolar power. This is discussed in section 4. Our conclusions are presented at the end of the paper.

Controlling volatility of wind-solar power in Germany
Normal passive buffers like pumped-storage plants with all their capacity limitations cannot alone control the volatility. Active buffers become necessary to guarantee a safe power delivery. To avoid CO 2 production, we choose windsolar power itself as active buffers. Assuming as above an already optimal distribution of wind-solar power devices across the nation, the additional windsolar power can again be expressed by a scaling factor, the strength α, and we get for the wind-solar energy 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 additional windsolar power is reduced as well enforcing a larger α value than expected from the average gain in power. To keep α within limits we apply the concept [7] of smart meters. Such devices control the electric consumption very effectively by setting higher consumption prices, when less power is available and lower prices, when there is a surplus of power. Smart meters act like passive buffering devices by moving the peaks of electric consumption to the peaks of wind-solar power [9] .
Of course a detailed simulation of smart meters is intricate [10]. However, we think that the following simulation of smart meters reproduces the basic effects satisfactorily, i.e. shows, how far the smart meter concept is applicable: has been defined as the energy of electric consumption. Now, if windsolar production has a surplus, it produces energy corresponding to a demand The smart meters now have the task, by decreasing prices for 1kW h to increase consumption and to achieve this E d (t ′ ). Clearly that is always possible -if necessary, due to exorbitantly low or even negative  371  372  373  374  375  376  377  378  379  380  381  382  383  384  385  386  387  388  389  390  391  392  393  394  395  396  397  398  399  400  401  402  403  404  405  406  407  408  409  410  411  412  413  414 Springer Nature 2021 L A T E X template Controlling volatility of wind-solar power in Germany 9  Table 2. τ = 0 implies: demanded power is delivered without delay, smart meters can pause for the moment.
prices. On the other hand, if wind-solar production is not sufficient, it produces energy corresponding to a demand E d (t ′ ) < E d (t) and t ′ < t. The smart meters have then the task, by increasing prices for 1kW h to decrease consumption and to achieve this E d (t ′ ). Clearly that is always possible -if necessary, due to exorbitantly high prices. Introducing the delay function τ (t) we write  Positive τ means that all power, not up to t but up to t + τ , has to be consumed at time t. As mentioned above, smart meters can achieve this by charging low prices. But there is a limit τ B beyond which prices must be unreasonably low or even negative in order to achieve consumption in advance up to t + τ. Therefore we require: To avoid τ > τ B electric power leading to τ > τ B is discarded as 'wasted' power and is removed from the system (see below). Since this 'wasted' power can be (nearly) arbitrarily high τ B becomes a perfect barrier for τ . But there is a limit −τ b for τ as well, below which the prices must be unreasonably high to enforce E d (t + τ ), τ < −τ b . Naturally τ b ≈ τ B , and for simplicity we set τ b = τ B . Therefore we also demand: In contrast to the barrier τ B there is no procedure to always keep τ above −τ B for any value τ B , since there is no unlimited power that we can put into the system. But we can form an optimized procedure for the delay function τ : Thanks to the installation of smart meters a quasi additional storage is obtained with its maximum given by  507  508  509  510  511  512  513  514  515  516  517  518  519  520  521  522  523  524  525  526  527  528  529  530  531  532  533  534  535  536  537  538  539  540  541  542  543  544  545  546  547  548  549  550  551 552 Controlling volatility of wind-solar power in Germany otherwise we move it back to τ B . In this way we get all the days, in which τ B is replaced by τ B0 . Computing τ (t) for the last configuration again delivers the days of the year with τ = 0 and therefore all days n σ with τ = 0. Only in the latter case the smart meters are active. A typical delay function τ is shown in  556  557  558  559  560  561  562  563  564  565  566  567  568  569  570  571  572  573  574  575  576  577  578  579  580  581  582  583  584  585  586  587  588  589  590  591  592  593  594  595  596  597  598 Springer Nature 2021 L A T E X template Controlling volatility of wind-solar power in Germany 13 incoming power has to be discarded to prevent increase of τ beyond τ B or τ B0 respectively.
The 'wasted' power arising by overflow of E s need not be small at all.
In fact, if all possible power is generated, the averaged power amounts to ≈ (1 + α) · 60GW (cf. table 1) and thus the average of 'wasted' power to ≈ α · 60GW. Getting rid of it directly is one way. This can be accomplished by reducing the wind-solar power generation, as soon as 'wasted' energy begins to build up. The advantage of this procedure would be, that the strain on the electricity network would not be essentially higher than for α = 0.
Exploiting this 'wasted' power for processes, e.g. for electrolytic and other chemical processes, would be an alternative. However, one has to keep in mind that the surplus power is really extremely volatile, as can be seen from Fig.3.
Apart from high peaks there are -more important -periods, even weeks, when there is no 'wasted' power available. 599  600  601  602  603  604  605  606  607  608  609  610  611  612  613  614  615  616  617  618  619  620  621  622  623  624  625  626  627  628  629  630  631  632  633  634  635  636  637  638  639  640  641  642  643 644 Controlling volatility of wind-solar power in Germany In our opinion the three characteristic numbers (n λ , n δ , n σ ) of the delay function τ are a realistic indicator for the volatility of the system. Having determined the domain, in which smart meters can be deployed, i.e. after fixing τ B , -in our case τ B = 3/2 days -the characteristic numbers (n λ , n δ , n σ ) can be determined from the power and load data as functions of the strength α and the storage capacity ς, cf. contribution. Instead an average demand of ≈ 60GW must be met. Following the surplus power approach this demand requires an average production of (1 + α) · 60GW . Therefore, the increase of the running costs is ws small → ws small · (1 + α) and the relative increase is given by α.
3 Importance of weak energy regimes, other options 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 windsolar power becomes important, once the wind-solar power production is weak, and therefore weak-wind turbines and solar cells with good performance in low light conditions will be essential for good surplus power production. Weakwind turbines having blades enlarged by a factor [14] √ β, greater height [15] and consequently higher wind speed enlarged by a factor [16] (γ) 1/3 , provide an increase of power generation by a factor of β · γ. We choose β · γ = 2. This doubles the surplus power production in the low-wind regime, P low = 2 · P .
In the high wind regime, however, the power production saturates, since these turbines have a reduced nominal power [16] P . This justifies the ansatz P low (t) = P nom · tanh (β · γ · P (t)/P nom ) , β · γ = 2 Weak-light performance of solar cells [17] depends on the material used [18].
Mono-crystalline PV modules [19], multi junction[20] with selected band gaps and in the future the new generations of DSSCs [21,22] may have good weak light performance. And we assume that with good weak light performance the generated power can increase -as in the wind power -by a factor of 2 in the weak-light regime too. (This approximation may be crude but is also less important 2 . So we choose for the total surplus power (here denoted as P low ) the ansatz P low (t) = P nom · tanh (2P v (t)/P nom ) , To demonstrate the importance of low energy production, we have selected a very low nominal power P nom for P low : P nom = P va · η Controlling volatility of wind-solar power in Germany The distinctly better outcome for the delay times τ is obvious in spite of the low nominal power P low . This emphasizes the importance of good performance in weak wind and low light situations.
We have no firm conclusions about the improvement of the results, when using offshore wind turbines. We have encouraging results for the year 2019 but for the year 2017 we have not got an improvement. As can be seen from Table 1 the contribution of offshore devices to the energy generation was still quite small in 2017. This may explain the controversial results.
Using solar cells as surplus power alone does not seem to be a good idea. over the year, whereas the total energy production amounts to ≈ 300GW , (averaged over the year). 80% consists of energy production on a fossil or gas basis for transport, warm water, space heating and process heating. Converting this non electric energy production into electric energy production should be possible, not completely, but to a large extent. 783  784  785  786  787  788  789  790  791  792  793  794  795  796  797  798  799  800  801  802  803  804  805  806  807  808  809  810  811  812  813  814  815  816  817  818  819  820  821  822  823  824  825  826  827 828 Controlling volatility of wind-solar power in Germany Therefore, the question is inescapable: Can all this electric power be generated by wind-solar power alone. Let us look at the consumption part first. The electric power curves of consumption did not differ much in the years 2015-20 and were characterized -apart from small waves due to the weekends -by large but slow changes on the summer-winter scale 3 . We think that this behavior is intrinsic and can be ascribed to the fact that there is no reason for most of the industry and private customers to drastically change their habits within days. Therefore we expect slowly changing consumption curves, when switching to electric power. And such curves represent minor difficulties. In contrast the volatile wind-solar power represents a major problem. But this part can be estimated by simple scaling as in section 2, leading to a scaling factor of 5. This means that the τ functions, their characteristic numbers and the domains, controlled by smart meters, essentially remain the same. But the number of devices and the storage capacities have to be multiplied by a factor of 5. According to our calculations in chapter 2 a storage capacity ς in the range of 1.5 − 5T W h will now be required. 4 Furthermore we can argue that in spite of its enormous volatility at least part of the now huge 'wasted' power can be used for chemical, in particular electrolytic processes, by which artificial fuel can be produced, e.g.
for airplanes. This would reduce the required wind-solar energy and the scaling factor.
Nevertheless, the huge power requirements represent an enormous challenge. Let us discuss the solar part first.
The scaling factor we have to use, requires a ratio between wind and solar power of ≈ 2 : 1 or 3 : 1. Thus the solar devices have to generate an averaged power (yet without smoothing) of nearly 100GW , and the question arises whether this is possible, since the capacity factor of solar cells is dismal for Germany [24,25]: about 10%. First numeric calculations dealing with this question presented unfavourable answers [26]. With continuously increasing power of computer codes taking into account higher levels of details, in particular structures of roofs [27] and facades [25] this question has now been answered convincingly in the affirmative. Smoothing requires another (averaged) ≈100 GW in our approach. Even that becomes possible. However, then nearly each roof and possibly part of the facades in Germany have to be covered with solar devices [25].
The needed number of wind turbines is enormous too. Their capacity factor amounts to [28] 25%, meaning that an averaged electric wind power of {200m}) are needed to produce this averaged power. But that is not enough, since the power must be controllable. In our approach this also leads to multiplying the number of wind turbines by a factor of (1+α), α ≈ 1.

Conclusions
Is it possible to switch all of Germany's present electric energy production to wind-solar power? The answer depends on how the enormous volatility of wind-solar power can be controlled. Our ansatz suggests marginalizing the formidable volatility by i) adding a substantial surplus of wind-solar power ii) installing smart meters 6 [7,8], iii) partly selecting different kinds of wind turbines and solar devices. iv) introducing a novel function describing this volatile system. The new function, in turn, depends on three characteristic Controlling volatility of wind-solar power in Germany numbers, which means, that the volatility of the system is characterized by those numbers.
When doing all this the results are encouraging: The electric storage capacity needed will be reduced to 0.3TWh -1TWh.
The prize to be paid will be a ≈ 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 power data [5] extend over the 6 years 2015 -2020, a period sufficient to show that wind-solar power is promising. And based on the present data, measured every 15 minutes during the period 2015 -2020, our approach avoiding excessive passive storage leads to the following conclusions: First, our approach is applicable to electric energy production in Germany as well as in other nations that do not have access to huge storage capacities. Second, our approach leads to the prediction, that Germany 's present electric power demand can be supplied by wind-solar power alone. Third, our approach does no longer exclude the hope that even if most of Germany's energy production switches to electric energy -which means an increase by a factor of about 5[2] -, this energy can be delivered by wind-solar power in a controlled fashion when applying our measures described above.
However, no matter how we slice it, the number of required solar cells and wind turbines will become tremendous: Only to satisfy at least the averaged consumption demand, nearly every second roof and possibly a significant part of facades has to be covered with solar cells. Moreover 650000 {170000} wind turbines of the 1.5M W {6MW} type (height 7 120m {200m}) become necessary.
But this arrangement, tremendous, as it is, succeeds only in setting equal the averaged consumption and the averaged wind-solar energy generation. The wild volatility is fully present and to control it requires a challenging effort -the subject of this paper. We offer a successful configuration that does not require much storage capacity. The price to be paid is to approximately double the already large number of wind turbines and solar panels. The running costs will equally rise by ≈ 100%.
The total number of devices will be so enormous 8 that the scenery of the landscapes will change 9 . Provided this fact is accepted by the public, windsolar power -according to the results of this paper -will have a realistic chance of becoming the leading generator of energy -in Germany and in other nations as well.