Solar Power, Revisited

Ten years ago, when we were building a large extension with reasonably good southern exposure onto our house, we considered adding solar panels to the roof.  Part of our motivation was ethical and aesthetic – we like the idea of being good stewards of the environment by reducing the pollution spewed into the air on our behalf; and there is an elegance, and a distinct emotional satisfaction, in making the best possible use of the limited resources available to solve a problem.

But we are also practical, and have no particular desire to play the martyr, so we also had our eye on the bottom line: how cost-effective would it be?  Alas, the answer, documented in a blog post at the time, was, “not very.”  Counting only the cost of the panels, themselves – ignoring the cost of ancillary equipment like static inverters and grid ties, ignoring the means and cost of energy storage to mediate the cyclical nature of the energy generation, ignoring the cost of installation, ignoring the cost of any financing required to manage the large up-front expense, and ignoring any maintenance costs, I estimated it would take about 26 years of savings on electricity purchases to break even.  Hence, we did not include a solar array in our construction.

By two years ago, however, the market for solar power installations had changed dramatically for several reasons:

  • The cost of the physical equipment has come down by a factor of 3 or more.
  • State laws requiring “net-metering” make it possible to use the public electrical grid as a “battery” to store excess energy during times of peak production and then to recover it on a 1:1 basis when your array is under-producing.
  • A new financing mechanism is available: a lease-to-own arrangement from companies like SunRun, SolarCity, Vivint Solar, and SunPower, (and perhaps others I’m not aware of or have forgotten to mention) in which a leasing company, taking advantage of their economies of scale and a store of working capital, pays the up-front costs and does the installation (at wholesale pricing, rather than retail pricing), tracks operation, performs any necessary maintenance, and guarantees a minimum amount of power production.

The net effect of those things is that using solar power is now not only cost effective but can actually, in many cases, be more cost-effective than not using it.

In response to that change in the market, we installed a lease-to-own solar array on our house and have been using it since then.  As I did with my original analysis of the proposed-but-never-built system ten years ago, I am now documenting our initial experience as solar power users.

First, a couple of caveats:

  • My experience, and my topic, is distributed (e.g. residential) solar installations. The economics of centralized commercial-scale solar power plants is very different.
  • My experience is based on a single system installed in rural New Hampshire. Although our roofline is 3 stories high and the tree-line is back at least 100 feet in the directions that matter, there is still some shading of the system, particularly in the winter when the sun angle is very low, and our seasonal variation in solar irradiance is very high.  A system in the desert of Arizona or California would behave differently.

 

Summary of System Performance

As of this month, the system has been online continuously for two years, so we have 2 years of data from which to assess its performance.

When I did my original analysis 10 years ago, I estimated we might average about 15% of the rated panel power output over the course of a year.  In truth, that number has been more like 11%; I presume at least some of the difference is the degree to which various panels are shaded by trees late in the day during the summer and for a large part of the day during the winter, which I didn’t account for in my original estimate.  Given the crude interpolation for expected solar insolation I used in my analysis (based on reported numbers for Boston, MA and Manchester, NH, both at least 40 miles away and in orthogonal directions), and on my simplification with regard to shading, I consider the 11% effective efficiency (relative to rated power) essentially in line with expectation.  I do note, however, that, had I used that number in my original analysis, it would have extended the estimated break-even time from 26 to 35 years; on the other hand, the increase in the price of electricity between then and now would have pushed that back down to 27 years.  Such are the vagaries of trying to make those kinds of estimates.

Constrained by available roof area and directionality, the surrounding vegetation, and the expected mean solar insolation given our relatively northern latitude and typical patterns of cloud cover, the array we installed was nominally sized to provide approximately 90% of our annualized electricity usage.  In practice, it turned out that the baseline years from which that average usage was estimated were somewhat forgiving in terms of weather, whereas the last two years were not.  Combined with a few transient lifestyle anomalies that increased our power usage, that meant the array, in truth, has provided for about 86% of our electricity needs since it was activated, despite the fact that is has performed pretty much exactly as predicted.  I expect in future years, with a bit of effort at reducing our energy usage, we can improve that to over 90% and might, as a stretch, even manage full break-even.

In terms of monthly cost, the original estimate projected a very modest savings by using the solar panel.  In practice, given the increase in energy usage and in the cost of electricity, the actual cost has been slightly above historical levels but not by a large amount.  Since the effective cost/kW-hr for lease of the solar system is below the current market rate for electricity, the net effect is that the increase we have experienced is not as large as it would have been without the solar array; I estimate we have saved about 6% overall on electricity costs in the past year as a result of augmenting grid power with solar power.

In summary, overall the system performed as expected: It provided on the order of 90% of our electricity needs for the year and saved us about 6% on our electricity bill.

And yet…

Notwithstanding the overall positive result we get from reviewing the annualized performance summary, there are several shortcomings in the actual day-to-day operation of the solar array that are worth some discussion because they illustrate the impact of available technology and public policy on solar power system performance.

 

Short-Term Variability

The following two figures show the power output from the array over the course of two representative days during the summer, when power output is nominally at its peak due to the long summer days and high sun angles.  The first represents a “good” day with clear skies and low humidity; the second, from two days later, represents a “bad” day with a complete overcast and rain.

Figure 1: Intra-Day Energy Production (Good Day) (July 5, 2017)

Figure 2: Intra-Day Energy Production (Bad Day) (July 7, 2017)

The first thing to note from these figures is that, even during the long days of high summer, the period of “high” power output only lasts for about 8 hours/day.  For four hours/day (or more), the power output is relatively small; and for a significant fraction of the day, the array does not generate enough power to supply the needs of a typical household.  Hence, a fundamental requirement for a practical solar installation is some mechanism for storing power during peak generation so that it can be used when the array is under-producing.  And, at the very least, that storage capacity must provide energy buffering on a time-scale of many hours.

The second thing to note from these figures is the variability in power output from day to day: the power output in the second figure is only a fraction of the power output from two days earlier.  That is, the day-to-day variations in power output can be quite large: the following figure shows the day-to-day fluctuations in power output, relative to the near-term ‘mean’ power output (represented by a 30-day running average) over the course of a year.

Figure 3: Variation in Daily Energy Production (relative to near-term mean)

 As shown, not only can the energy output vary dramatically from one day to the next, but there are many extended periods of very low or very high energy output (for example, observe the low-output region near day 50 that extends for something like 15 days).  As a result, any energy storage mechanisms provided to buffer the variations in energy production must be able to buffer not only intra-day variations, that last for hours, but also extended periods of production surplus/shortfall lasting days at a time.

 

Seasonal Variability

In addition to intra-day cycles and periods of weather-related multi-day surpluses/shortfalls, there are also large seasonal variations in energy production.  The following figure shows daily the energy production over the course of a year, normalized to the annualized average.

Figure 4: Daily Energy Production (Normalized)

As we can see, production varies dramatically over the seasons and is near zero during the winter months of short days and low sun angle; there are only two short periods, just at the vernal equinox and shortly after the autumnal equinox, for which the actual  panel production roughly matches its annualized mean.

The following figures show two views for a comparison of panel energy output to our household energy usage (estimated from monthly energy bills from the local utility).  Again, for the vast majority of the year, the panels are either significantly under-producing or significantly over-producing relative to the actual demand, which means that, for the vast majority of the year, system operation will depend on long-term storage capacity with a duration of months, rather than days or weeks.

Figure 5: Daily Energy Production vs. Usage (Normalized)

Figure 6: Ratio of Daily Energy Production to Daily Energy Usage

Overall, ignoring short-term variations and focusing only on the seasonal changes, there are something on the order of 200 days during the year in which the system is dependent on some amount of grid power availability to make up for the shortfall between energy consumed and energy generated by the solar array.  On the other hand, during the remaining 165 days of the year, the array is generating more energy than is required for immediate consumption – nearly twice as much in the months immediately following the vernal equinox – and that energy must be diverted into some kind of long-term storage if it is not to be wasted.

 

Utility as Battery…

There are no energy storage technologies, existing or envisioned to be available in the near future, that are capable, at anything remotely approaching an economically efficient price or a physically useful form-factor, of providing storage on the seasonal time-frame required to buffer the ups and downs of localized on-site solar energy generators.  Instead, the standard solution to the “storage problem” is a matter of public policy, not of technology: “net metering” or some similar arrangement.

Net metering allows distributed solar power sources to use the local electrical grid as their “storage device” by dumping their excess power into the grid when they are over-producing, and drawing make-up power from the grid when they are under-producing.  Of course, the grid doesn’t actually “store” the power.  Instead, when solar arrays are dumping excess power into the grid, the other generators providing power to the grid reduce their output to keep overall power generation matched to overall power consumption.  When solar arrays are under-producing, the other generators ramp their output back up to compensate.

In other words, net metering allows local solar power arrays to act as small-scale generators, feeding power into the grid as it is available; and power from the solar arrays is always used in preference to other generating sources – solar arrays are always allowed to dump as much power as they can produce into the grid, while more traditional power plants are required to throttle down their output to compensate.

From an environmental standpoint, that policy makes great sense: “clean” energy sources, which consume no non-renewable fuels and emit no pollutants, are always used to replace “dirty” sources when they are available; and the “dirty” energy sources make up the difference only when the “clean” sources can’t keep up with demand.

But that policy has some limitations, both logistical and economic, which mean, at least as long as we use net metering as a substitute for a real solution to the storage problem, that the use of solar power – and other intermittent generation sources – still depends heavily on some kind of legally-enforced public subsidy to remain economically viable; and the fraction of power that can be provided by such sources will be limited by the amount of power the grid can absorb without saturation.

 

…But It’s Not Really a Battery

The fundamental logistical limitation of net metering is that it can only work when there are few enough solar generators on the grid that they cannot keep up with the overall demand even during their peak output times.  In that case, there are always other generators supplying power to the grid that can cut back on their output.

But what happens if the bright solar future everyone dreams of actually comes to pass?

During the times of year that allow for peak solar output, and in the middle of the day when output is at its highest, a solar array might be instantaneously generating 4 to 6 times as much energy as is required to sustain its local load.  Aggregated over the entire grid, that means that, if solar arrays designed to supply more than about 15% to 25% of average demand were online and operating at peak output, there would not be enough instantaneous demand on the grid to absorb the excess power they were generating.  The other generators could shut down completely, but there would still be more electricity being dumped into the grid by solar arrays than was being drawn out of the grid by consumers.  With no place to dump the excess power, it would simply be wasted.

That would seem to be a high-grade problem – we are generating so much solar power that we have to throw it away!  The problem is that the wasted power really does just go away.  It isn’t accumulated and returned to the grid later, when there is a deficit; and, hence, it drives up the effective “net metered” per-unit price of solar energy for everyone with a solar array because the wasted energy is not included in the net metering tally – you don’t get to draw down a credit for power that you dumped on the ground instead of transferring to the grid.  Because that energy is being thrown away, the solar arrays generating that power are, in effect, operating only at some fraction of their nominal capacity instead of at full capacity.  And, hence, beyond that point of diminishing returns, new units of solar power added to the grid do not displace non-solar generators on a 1:1 basis but only at some fraction of that, a fraction that becomes increasingly smaller for every new unit of added solar power.  The more solar arrays are added to the grid, the less cost effective solar power becomes.  That implies that the economics define an upper bound on the fraction of overall power that can be supplied by solar installations if we are depending on net metering to solve the storage problem.

Moreover, in the dead of winter and at night and during stormy weather, when solar arrays are generating very little relative to overall demand, grid customers – including those with solar arrays which are nominally “net positive”, who produce more energy than they consume — still rely almost entirely on the availability of traditional power-plants to provide the needed power.  But the economics of power generation – the cost of building and operating a power plant – are predicated on the power plant being able to generate revenue by selling some significant fraction of its rated power output over its lifetime.

If a significant fraction of the overall power demand is supplied by solar arrays during sunny times, those non-solar generators will not be economical to build and operate merely as a backup source, to supply power during the dark times, unless they can charge significantly more for the power they do generate.  In other words, to ensure that the generating capacity remains available to provide power during the night and stormy weather and the winter, the price we pay for that electricity will need to increase substantially.  Hence, despite the fact that more of our power may be provided by “free” solar power arrays, the overall cost of electricity will probably not fall, and may even increase, in order to ensure that the non-solar generation capacity remains available to fill the gaps.  That doesn’t eliminate the environmental benefits of replacing carbon-based generation with solar generation, but it does mean that we may not see any significant savings in overall energy costs as a result.

 

Subsidized Virtue

Beyond those long-term logistical issues, in the short term net metering creates a purely economic problem, one that is the source of much of the opposition to the net metering policy.

For the moment, a large part of what makes solar power generation cost-competitive with other energy sources for consumers is the fact that power dumped into the grid and power returned from the grid under net metering rules are treated symmetrically.  When I dump excess power from my solar array into the grid, I accrue a credit for that much power; when I draw power back out of the grid to make up a deficit, I draw down that credit.  Hence, every unit of power my solar array generates not only replaces exactly one unit of power that I would otherwise have purchased from the power company but also saves me all the money I would otherwise have paid for that power.  As long as the amortized per/kW-hr cost of my solar array is not more than what I saved on the electricity I didn’t need to buy, I am happy.

From the standpoint of an electric company, power they provide to me to make up my credit is power they otherwise would have sold to me at the retail price; and, hence, when I build up that credit in the first place, by sending them my excess power, they are, in effect, paying me the retail price for that electricity.

But, the electricity that my power displaces when I am building up that credit – the power no longer provided by other generating sources – is power that the electric company would otherwise have purchased at the wholesale price.  In other words, net metering requires the power company to buy my solar power at the retail price instead of buying that power from some other generator at the wholesale price.  Every unit of solar power I send to the grid under net metering, therefore, results in a loss of revenue to the power company equal to the spread between the wholesale and retail prices of electricity.

Since power companies are regulated monopolies, guaranteed a “reasonable” rate of return by the Public Utilities Commissions that set their prices, the companies make up that revenue shortfall by increasing the price of power overall – not only for me but for all the other rate-payers.  That means, in effect, that all the other utility customers subsidize my use of a solar power system – the 6% I saved on electricity last year by drawing power from my solar array, and some significant fraction of my annual cost for installation and maintenance of that array, was paid for by all the other people who buy electricity from the grid.  And the more solar power systems are added to the grid, the more those other electricity customers have to pay to subsidize them.

People who understand they are paying for other peoples’ solar aspirations are, understandably, unhappy about it.  That is a primary reason why the amount of net-metered power that may be supplied to the grid and/or the size of solar arrays that may feed power to the grid under net metering rules is strictly limited, and why each proposal to increase those limits is resisted with such vigor.

 

Sanity Check: A Crude Cost Analysis

To sanity-check the contention that the affordability of solar energy depends on a public subsidy, I provide a quick and crude analysis of the effective production cost for electrical generation given various fixed and variable costs for different types of generators.

I have chosen to use a modern gas-powered commercial generator as my stand-in for “traditional” generation sources, and my analysis includes a fundamental assumption that some minimum amount of traditional (non-solar) generating capacity must be in-place (and paid for) in order to provide peak load power when the solar cells are not operating at peak capacity.  Hence, the overall cost of electricity in this analysis includes the cost not only of enough non-solar capacity to carry the average residual load but enough capacity to carry the peak residual load.

To keep things simple, I have also opted to ignore the effect of grid saturation: I have assumed that the grid can accept and (somehow) store/redistribute all the energy provided by solar power systems and will not simply dump energy on the ground when those systems are providing more power than the grid needs.  In the event that power from grid-attached solar systems were to be, in fact, wasted, it would make my analysis results look worse as more solar arrays were connected to the grid and more power went unused.  Hence, these results may be considered somewhat “conservative”, in the sense that they put solar power into the best possible light.

For purposes of this analysis I have used the following cost estimates (sources cited):

tana = 48 yr

Time-frame over which to amortize fixed costs; based on the average retirement age for gas power plants per powermag.com (“America’s Aging Generation Fleet”, Neil Powell, 1/28/2013),pulled 08/05/2018.

Cf.gen = $812/kW

Fixed cost for construction of a modern gas power plant; 2015 estimate from eia.gov, pulled 08/05/2018

Cv.gen = $0.03019/kW-hr

Variable (per-kW-hr) operating cost for a traditional gas power plant; 2016 estimate from eia.gov, pulled 08/05/2018

Cf.sol = $24600/kW

Fixed cost for construction and maintenance of a residential solar photovoltaic array; based on the amortized leasing cost of my solar array over its 20 year lifetime (which includes design/installation, financing, and maintenance) and deducting a 20% allowance for leasing company profit.  Note that, based on the “pre-purchase” option in the contract, the long-term financing appears to represent about 1/3 of the overall cost.  This estimate probably understates the maintenance cost, since it includes only 20 years of maintenance while we are amortizing over a 48 year period.  Also, we don’t have enough experience with modern solar arrays to know for certain that we can realistically expect a 48 year lifespan from them, so this may under-state the cost for a plant with that long a lifetime if it must be replaced earlier.

Note that eia.gov (pulled 08/05/2018)  estimates the cost of a commercial solar photovoltaic generating plant (average size = 10 MW) as about an order of magnitude less that the estimate I am using (construction only; no maintenance).  That is probably due not only to economies of scale involved in producing a commercial plant but also to the fact that such plants may be sited and designed specifically to optimize captured solar insolation, rather than conforming to the pre-existing architectural and siting constraints of a residential retrofit (e.g. they can site such plants in the Arizona desert, instead of in tree-covered New England, and use sun-tracking panels to maximize capture instead of using existing roof geometries and pointing angles).  Since I am focusing on the effects of distributed (i.e. residential) solar installations, I have chosen to use an estimate appropriate for that.  Commercial-sized solar generation is a different beast, entirely.

For comparison, if you simply extrapolated from the retail prices of a typical solar panel (e.g. $1488 for 4 265W panels @11% effective efficiency, with no ancillary equipment, installation, maintenance, or financing; Home Depot online, 08/05/2018), the estimated cost/kW would be more like ~$12,760/kW.

Cv.sol = $0.00000/kW-hr

Variable (per-kW-hr) operating cost for a solar array; since I included maintenance as part of the fixed cost, I presume the per-unit operating cost is zero.

Eannual = 3,762,462 GW-hr/yr

Total annual electricity consumption in America; 2016 estimate from eia.gov, pulled 08/05/2018

Epeak = 1036.342 GW

Peak load capacity in America; 2016 estimate from eia.gov, pulled 08/05/2018; this is actual peak load capacity, but I assume this is also how much peak load capacity is actually required.

fpeak.sol = 25%

Fraction of mean solar array capacity that is available for use in servicing the peak load.  This is a crude estimate, based on an assumption that peak loads typically occur in the late afternoon/evening on hot summer days, when the solar array efficiency has begun to drop off due to low sun angle.  See figure 1, which shows that, on the best summer days, the array output has dropped to about 20% of its peak output by 6 in the evening.  Of all the assumptions in this analysis, this is the one I define with the least confidence.  On the other hand, as we will see in the resulting graphs, its effect is relatively small compared to the effect the initial capital expense for building the arrays.

fsol

Fraction of total electrical power provided by distributed solar arrays.

From these definitions, we can estimate the effective cost of electricity given that some fraction of the average power load is provided by solar arrays:

 

Baseline (no solar power generation):

Etotal = Eannual x tana

Total energy used during the analysis period

Pbase = [ Cf.gen x Epeak ] / Etotal + Cv.gen

Pbase = $0.0349/kW-hr

Effective production cost of electricity generated entirely by commercial gas generators

 

With solar power generation:

Epeak.gen(fsol) = 1 – fpeak.sol x fsol

Amount of peak capacity that must be provided by non-solar sources, as a function of solar generation capacity

Psol(fsol) = [ Cf.gen x Epeak.gen ] / Etotal + Cv.gen x ( 1 – fsol )

+ [ Cf.sol x fsol x Eannual ] / Etotal + Cv.sol x fsol

Psol(1%) = $0.0351/kW-hr

Effective production cost of electricity with 1% of capacity provided by solar power

Psol(10%) = $0.0376/kW-hr

Effective production cost of electricity with 10% of capacity provided by solar power

Psol(25%) = $0.0416/kW-hr

Effective production cost of electricity with 25% of capacity provided by solar power

 

As we can see, given our assumptions about cost and peak load generation, the effective cost of electricity rises as the amount provided by distributed solar power increases.  That is due to two factors:

  • The much higher fixed cost of distributed solar power installations relative to commercial-scale gas generating facilities
  • The need to retain a significant amount of non-solar generating capacity to handle peak power loads.

Of those two, by far the largest influence is the fixed cost of the solar power systems.  The following figure shows the effective cost of electricity, as a percentage of the base cost, as the amount of residential solar generation capacity on the grid increases; the red (dashed) line illustrates the effect of assuming that the solar arrays can provide 100% of peak capacity (so no non-solar generation is required):

Figure 7: Cost of Electricity as a Function of Residential Solar Capacity

To be fair, I must note that the equivalent graph for solar capacity provided by commercial-scale solar installations – which (as noted above) are presumed to be much less expensive to build on a per-unit basis – reverses the trend; for such commercial-scale plants, increasing the use of solar power decreases overall electricity costs because the higher infrastructure cost for solar arrays does not overwhelm the savings in per-unit energy generation:

Cf.sol = $2921/kW

Fixed cost for construction and maintenance of a commercial solar photovoltaic array; from eia.gov (pulled 08/05/2018).

Figure 8: Cost of Electricity as a Function of Commercial Solar Capacity

Given my assumptions, the break-even point – the point at which adding new solar capacity has no net effect on electricity cost – is at about $13,000/kW (amortized over a 48 year lifetime) for solar array installation and maintenance.  If the cost falls below that, solar power is unambiguously a more cost-effective generation source provided that all the arrays can operate at full capacity without saturating the grid.  If, as described above, the total instantaneous generating capacity of the solar arrays during peak operating times exceeds the load on the grid, then maintaining full-capacity operation would require some form of long-term storage on the grid to absorb that power while it was being generated and then to release it when it was needed.

© Copyright 2018, Augustus P. Lowell

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