The Final Nail in The Coffin Of “Renewable” Energy

My work deals in very formal proof, by taking actual demand hour by hour, and actual wind and solar generation ditto, and then calculating what capacity can be absorbed without generating a surplus, what combination of wind and solar produces the least requirement for storage given a particular round trip efficiency, etc. using linear algebra.

Formally, let G be a matrix of hourly generation capacity factors (derived from actual hourly production by dividing by the nominal capacity) with a column for wind and a column for solar. Let D be a vector of hourly demand. Let C be a 2×1 vector of nominal capacities to be found subject to the constraints we impose on the problem.

We can write that the vector of hourly surpluses and deficits S is

S = C.G – D

If we require that S be strictly <=0 for every hour we have a locus of solutions for C giving the maximum combinations that are consistent with no surplus generation. S is then the hourly backup dispatchable generation required to meet demand. Elements that are constraining will meet the condition C.Gt=Dt where t is the time subscript. High levels of supply (high Gt capacity factors for individual hours) may or may not coincide with low levels of demand, but in the long run we can expect that will occur. So we are looking in practice at the fleet maximum generation for wind in the low demand small hours when there is no solar, which is say ~80% of nominal capacity, plus the amount of solar given that level of wind capacity which does not result in a surplus on sunny summer Sundays, at least for grids where both can contribute sensibly. The LP will find the solutions which can also be found by iterative approximation.

If we now allow for storage, we can increase the values of C to generate surpluses for some individual hours. If we partition S between hours of surplus and hours of deficit we can calculate additions and and withdrawals from storage.

Where there is a surplus we can record it as contributing St (surplus for hour t) multiplied by the efficiency of the storage process to the store – say 85% for pumped storage pumping, 60% for a PEM electrolyser or 90% for an inverter feeding a battery. These amounts may be constrained by available capacity in MW for pumping, electrolysis or inverters, resulting in wasted surplus energy. This becomes an important consideration, since the volume of surplus is highly variable, and it will never be economic to provide capacity to absorb the largest surpluses that occur only rarely when high output and low demand coincide. However, and initial calculation can be run on an unconstrained basis which permits a storage process plant duration curve to be produced to input to the economics of capacity. This chart shows some sample surplus duration curves for the UK, illustrating the difficulties of making hydrogen electrolysis an economic proposition.

https://datawrapper.dwcdn.net/nZM72/1/

Where there is a deficit we can proceed similarly: any deficit in excess of the redelivery capacity of the storage must be met by dispatchable backup. Otherwise the redelivery volume Vt reduces the store by an amount Vt divided by the efficiency of redelivery – say 95% for pumped storage hydro, 50% for a hydrogen fuelled CCGT operated intermittently, 90% for a battery inverter. Finesse the calculation by allowing for leakage from the store over time, including energy use for cooling batteries etc. ad lib. Alternatively, simplify by using the round trip efficiency on one or other leg.

By cumulating the surpluses and deficits we can derive the volume in storage. It is convenient to calculate the volume of storage required in MWh (or in reality, the significant number of TWh required) as the difference between the maximum and the minimum of the volume in storage.

As a starting point, and assuming we are dealing with data covering n years, (n integer, >=1) it makes sense to constrain the solution such that the volume in storage at the end of the period is equal to the initial volume. By optimisation we can find the combination of wind and solar that minimises the maximum storage requirement while incurring no wastage beyond round trip losses through storage. We should not be assuming that there is a stock in store for free that doesn’t need to be replenished. This chart shows the storage profile for hydrogen or a 75% round trip (pumped storage, battery) store over the course of a year for GB.

https://datawrapper.dwcdn.net/ZmrQw/1/

In practice, and particularly since in the real world no year is exactly like another, it makes sense to explore the sensitivities around such a solution to see how relatively robust it is, and to consider the tradeoffs between different ratios of wind to solar, and between storage and additional renewables capacity with associated wastage, and using dispatchable backup not fuelled from the storage. It is important in conducting the analysis to do so on a marginal basis to understand how the tradeoffs actually perform. We can also analyse the maximum flows required to fill storage and meet demand, and hence begin to examine the need for extra transmission capacity.

Multiyear analyses can reveal huge differences in storage and capacity requirements or potentially largely useless surpluses depending on inter year variablility of renewables performance. I have run some using over 35 year runs of refactored weather data at hourly resolution – over 300,000 rows. Covering that 1 in 35 year bad year can be tough, especially if it comes in a run of poor years. A “good” renewables year will throw up a lot of wastage if we have covered the bad year properly.

The reality is that these calculations require spreadsheets of thousands or even hundreds of thousands of rows (and good quality input data), but the actual calculations are not particularly difficult to set up, and the solutions can often be found using Goal Seek or Solver options, with iterative interpolation as a fallback because of the large number of implicit constraints. They certainly cannot be done on the back of an envelope, with the possible exception of the simplified Thursday Island example which mainly considers monthly data that makes it easy to appreciate the nature of the calculations and the importance of seasonality (even 10 degrees South of the Equator in the Torres Strait). I would encourage you to read it: I put it together having been inspired by the late Roger Andrews, who wrote a good number of articles looking at renewables intermittency and performance at Euan Mearns’ site, performing similar calculations.

It is instructive to consider some basic mathematics. Let g(x) be a capacity factor duration function, defining the proportion of capacity that is produced less than x proportion of the time. It is analytically convenient to consider functions g(x) of the form g(x)=x^n. These have the property that integral g(x) from 0 to 1 is 1/(n+1), and is the average capacity factor. Thus for n=1, g(x)=x, and average capacity factor is 1/2, or 50%; for n=2 g(x)=x^2, and average capacity factor is 1/3 or ~33%, etc. They are actually quite good approximations to real world duration curves for fleets of wind farms, although they need scaling by a fudge factor that reflects that wind is never consistent enough to achieve full capacity across a fleet of farms, but tops out at around 80% of capacity – see the recent wind record in the UK at 20.918GW plus some curtailment cf capacity of 28.5GW.

For a capacity of C, the duration curve is simply Cg(x). For demand we should also properly consider duration curves, recognising the several hours of low demand overnight, rising towards a plateau during much of the day before the early evening peak, subsiding back again to overnight levels, amplified by the weekday/weekend and seasonal factors. For simplicity I will look at convolution with baseload and peak levels of demand, respectively B and P. If the proportion of hours that are baseload is α the average demand is αB + (1-α)P. A fuller analysis looks at a full convolution integral with the different levels of demand. We shall assume that there is no correlation between wind output and time of day which is reasonable for the UK, but not true in e.g. Australia where it tends to be windier at night – or seasonally (whereas in the UK it tends to be windier on average in winter). B and P can also be thought of as low season and high season demand rather than merely diurnal fluctuations.

Following Pollock’s principles we have that all curtailment is avoided when C<=B. For C>B we expect curtailment. The expected rate of curtailment is Cg(x)-B over the interval between x:g(x)=B/C (or x=(B/C)^(1/n) ) and x=1, and the total curtailment is the integral of Cg(x)-B over that interval. Integrating the function we get (Cx^(n+1))/(n+1)-Bx which evaluates to C-B at the upper limit and C(B/C)^((n+1)/n)/(n+1)-B(B/C)^(1/n) at the lower limit, and this level of curtailment applies the proportion α of the time. While C<=P that is the source of curtailment. For C>P we have a similar analysis applying to the peak hours, applying for (1-α) as a proportion of the time. The analysis can be extended to more levels of demand in an obvious manner, and in the limit as a convolution.
For n=1: we get curtailment of
α(C-B)(1-B/C)/2 for C>B, plus a further (1-α)(C-P)(1-P/C)/2 for C>P
Marginal curtailment for an increase in capacity is found by differentiating with respect to C.
α(1-(B^2/C^2))/2, plus a further (1-α)(1-(P^2/C^2)/2 for C>P.
Marginal production is ½, and the useful production, U is ½ less marginal curtailment. The effective cost of marginal capacity is then the LCOE multiplied by 1/2U to allow for the wastage. Capacity additions are asymptotically useless as B/C and P/C tend towards 0 with increasing C. However, for values of C that are only slightly above B marginal curtailment is small, and only occur in baseload hours. If we set C equal to average demand so that total generation by wind is C/2, equal to average capacity factor multiplied by average demand – the Pollock criterion – we find that B<C<P so we are only concerned with curtailment during baseload hours, but that curtailment is positive. There are no obvious grounds for stating that it is optimal. However we have the basis for a cost curve relative to levels of grid penetration (though we should add in other costs caused elsewhere in the system to make a proper tally: already by the time there is sufficient capacity to cause curtailment there are plenty of costs for extra transmission and grid stabilisation and costs imposed on other generators as I have outlined already). An optimised grid would see marginal long run costs of different kinds of generation equal if you can define those long run costs. In practice, the optimisation would need to be risk based, reflecting probabilities for many different parameters.

We can replace B in our formulae by a demand function D(t) where 0<t<1 (year), and integrate over t to provide a convolution.

Higher values of n (including fractional values) produce more complicated expressions for curtailment, but for given C, curtailment occurs less of the time, and there is more headroom to increase capacity at little curtailment penalty. In practice, empirical capacity factor duration functions may have an interval of low to zero output and never attain a 100% capacity factor at the fleet level. Modifying the form of g(x) accordingly may provide more realistic answers, but the underlying analysis gives a feel for how the variables interact to influence the tolerable level of wind generation. Comparison with empirically determined surplus duration curves such as these

https://datawrapper.dwcdn.net/nZM72/1/

goes some way to validating the approach. In turn, they highlight the problems of trying to create an economically rational storage system as an alternative to dispatchable backup. That is perhaps the real issue here, especially when combined with the economics of storage turnover.