Contents

List of Figures

List of Tables

The fundamental unit of energy is the Joule. One Joule is the energy needed to push a force of one newton though a distance of one meter. One newton is the force necessary to uniformly accelerate a mass of one kg from rest to a velocity of one meter per second in one second. One kilogram is the mass of one liter (1000 cc) of water.

Heat is a form of energy. One calorie is the amount of heat needed to raise the temperature of one gram (1 cc) of water one degree C. At sealevel pure water freezes at 0 Celsius (Centigrade) and boils at 100 Celcius. There are 4.19 Joules in one calorie.

The British Thermal Unit is another unit of energy. One BTU is the amount of energy needed to raise the temperature of one pound (454 g) of water 1 degree Fahrenheit, or 5/9 degree C. One BTU = 1 lbF x 1 cal/gC x 5/9 C/F x 454 g/lb x 4.19 J/cal = 1056.8 J $\approx$ 1055 Joules, the official conversion number. One Quad is one quadrillion BTU – that’s $10^{15}$ BTU or one PBTU. A Quad is 2.93x$10^{11}$ kWh or 293 TWh. The United States generates about 4 PWh electric energy each year, or 13.65 Quad. ConvertWorld.com has a handy energy units converter.

Energy and work are the same thing. Power is different. Power is the rate energy is expended, or how fast you do work. The standard unit of power is the Watt. One watt is one Joule expended each second: W = J/s. We can get units of energy back again by multiplying different numbers of Watts by different lengths of time. The unit we see on our domestic electric bill is the killowatt-hour. 1 kWh = 1 J/s * 1000 * 1h x 3600 s/h = 3.6 million Joules, or 3.6 MJ. A kWh is easy to conceptualize: it’s the amount of heat given off by a good table toaster burning continuously for one hour. One does not usually generate or consume energy at a constant rate (constant power). Usually one consumes a little here, a little more there, then turns in for the night and consumes a bit more in the morning. They all add up: energy is the integral of power over time:

$E={\int \; <!--nolimits-->}_{t_0}^{t_f}P\left(t\right)dt$

Something to keep in mind: in energy discussions it is common practice to speak in terms of MegaWatt (MW) capacity (one million watts) or GigaWatt (GW, one billion watts.) MW and GW are context-sensitive units of power, and mean different things to different people at different times. A kWh, on the other hand, is a unit of useful work and means just one thing: a kWh is a measure of how much energy you consumed at a time you needed to use it. Not some other energy at some other time. Your power company charges you for the energy they deliver to you at the time you need to use it.

Electric utilities call this “keeping the lights on.”

1.1 Three-Part View of Power Generation

Figure 1 is a simplified cartoon of what power utilities are up against when trying to keep the lights on, and broadly illustrates the three general ranges of load fluctuation.

1.2 Generating Capacity, Capacity factor, and Dispatchability

The rated or “nameplate” capacity of a power plant $P_r$ is the maximum operational power the plant is designed to continously deliver. Modern thermal plants (coal, nuclear) may have $P_r$ of one GW or higher. Operating in baseload mode, they may deliver $P_r$ for months at a time, but not indefinitely. Parts wear out. Coal flues must be cleaned. Most nuclear plants must be shut down to be refueled every other year or so, which can take six to eight weeks. Over time the average power output $P_a$ of such a plant is less than the rated design power $P_r$, and the ratio of the two, $P_a∕P_r$, is called the plant’s Capacity Factor. It is sometimes useful to rigorously define the capacity factor or (plant factor) over a specific time interval $\left[t_0,t_f\right]$ where $t_f-t_0$ is usually over a year and may span several decades:

Operating in baseload mode, coal plants can achieve capacity factors of over 95%, vs. 90 - 92% for nuclear. But coal, and Pulverized Coal (PC) in particular, is also operated as load-following plant, which lowers the overall capacity factor of U.S coal to about 64%. While many nuclear plants may be operated as load-followers as well, their high initial construction and finance costs and low fuel cost encourage baseload use as much as possible.

Natural gas plants come in several varieties. The simplest are open-cycle gas turbines (OCGT) wherein hot combustion gas drives a series of one or more gas turbines and is then exhausted to the atmosphere. Modern pulverized coal and open-cycle natural gas plants may achieve thermal efficiencies of 40%: four tenths of the thermal energy in the natural gas is converted to electrical energy for the grid. The rest goes up the flue. Combined-cycle gas turbines (CCGT) add a water boiler after the gas turbines to use remaining heat in the exhaust gas to generate steam for subsequent steam turbines. Combined-cycle plants may reach thermal efficiencies of 60%. Either way, the gas turbine is well-suited to peak-load operation, and open-cycle plants are usually used in this mode. Capacity factors may be 20 - 40%.

For intermediate load-following – and peaker as well – it’s hard to beat a dam-backed hydroelectric plant if one is available. The water turbines at Grand Coulee and Hoover dams can spin up from near stop to full power in less than two minutes. Grand Coulee has a nameplate capacity of 6.8 GW, and generated 21 billion kWh useable electricity in 2008. With 8766 h/y, the Grand Coulee system, given enough water, could have generated 6.8 GW x 1 y x 8766 h/y = 59.6 billion kWh, so it’s plant (capacity) factor was 21/59.6 = 35%. With lower total flow and higher head, Glen Canyon and Hoover dams operate with capacity factors of about 24%.

Dispatchability

It is important to realize that thermal plants – coal, gas, and nuclear – and dam-backed hydro, are all dispatchable technologies. Baring planned (or unplanned) shutdowns, they are always available to deliver the amount of power their operators request (up to $P_r$) when their operators schedule them. Their power is dispatched as needed. That anemic-looking 24% plant factors for Glen Canyon and Hoover is actually extremely useful to the Los Angeles and Las Vegas electric companies needing to supply peak power during the late-afternoon summer maximum draws. Hoover dam can’t supply its full 2 GW rated peak for more than a few days (or weeks) at a stretch, but with foresight and planning its 500 MW average may be dispatched along with baseload coal and nuclear, and intermittent solar and wind, to deliver very reliable power to the grid exactly when the grid customers (you and I) need it. A 24% plant factor or capacity factor for a dispatchable load-following plant has a very specific and very useful meaning.

Capacity factors for intermittent renewables such as wind and solar photovoltaic mean something as well. But in the absence of storage, what intermittent capacity factors means is something quite different than the same term and numbers for dispatchable thermal and hydro.

Wind and solar are non-dispatchable. They are highly available whenever the wind blows and the sun shines – but that is not necessarily when the grid operator most needs them. A modern wind turbine may have a peak capacity of 2 MW, and be sold as a “2 MW capacity” turbine at a rated output wind speed of perhaps 10 m/s: it will deliver 2MW only for wind speeds between 10 and 25 m/s (its cut-out speed), and experiences a smooth decrease in output power as wind speed falls beneath 10 m/s as illustrated in Figure 2.

The capacity factor for non-dispatchable resources is defined as for dispatchable resources:
$C_f$ = (Actual energy out) / (theoretical maximum energy out) as measured over some extended interval of time at the installed location. Due to seasonal variability, the time interval is typically a year or more for wind systems.

At our present low levels of penetration ($\sim$ 3.5% in the United States), our grids and subsidies are set up to absorb all the electric energy each turbine can generate. The resulting capacity factors then reflect only the variability in wind speed, and of course varies with location. Capacity factors currently average about 33% for United States onshore wind. This means a typical 2 MW turbine actually delivers an average of 670 kW over a year of operation, that is the average maximum it can generate. The resulting kWh of (often) useful work are delivered when the wind blows, not necessarily when the grid operator (and his customers) would most like to have them.¹

It bears emphasis: the nameplate capacity of any power generator is relevant only within the context of its capacity factor, dispatchability, and intended use.

2.1 Marginal Utility

Table 1 details U.S. electricity generation by source. Figure 3 shows the United States currently generates about 4 trillion kwh or 4 PWh of electric energy each year, of which 3.46% comes from onshore wind. With natural gas providing 25% of our total power, it is easy to smooth out the load fluctuations introduced by wind intermittency,and save natural gas fuel in the process. The incremental or marginal cost of adding a wind turbine to the grid at such small penetration is essentially just the capital cost for the turbine and any needed transmission, plus ongoing operating and maintenance (O&M) costs.

However, if one wished to power an entire grid from wind, one needs to consider both the wind capacity factor and the actual wind patterns within the areal extent of the grid as well. For example, if one were living in an area where wind turbine capacity factors average 33%, one might simply say: “Well, if on average the wind blows only a third of the time, then I just buy three times as many wind turbines and I’m golden.”

Or not. Suppose your grid needed to supply 1 PWh over a year’s time. There are 8766 h/y, so 1PWh = 114 GWy. Again assuming 2MW turbines, you might naively think you’d just need
114GWy/(2MW nameplate watts/turbine * 0.33 obtainable watts/nameplate watt) = 171,000 turbines. Which would be true and that many turbines would deliver the desired 1 PWh over the course of the year.

Oh, wait... which PWh did you say? Desired. Needed. When you and I need the power. Not whenever the wind might decide it can deliver it. As illustrated in Figure 1 above, you and I don’t consume a steady constant 114 GW over a year. Our load fluctuates. Likewise the wind power over any grid (alas, of any size) fluctuates as well, and the two fluctuations rarely match. There will invariably be times when there is not enough wind available to meet electric demand, which must either then be curtailed, or be met by backup and storage. Likewise there will be times when there is more wind available than the grid can absorb, even while replenishing storage. Then some turbines must be feathered, and the overall capacity factor will be reduced beneath its nominal 33%.

Even if you (and your ratepayers) could live with even further reduced efficiency, overbuilding enough wind to meet peak load rather than average isn’t going to help much: it would help only if the wind pattern at one turbine (or farm, or region) did not correlate with that at the next, but we all know that it does. In a recent study of the United States’ PJM Interconnect region (Pennsylvania, New Jersey, Maryland & friends), Budischak et al. found wind patterns would positively correlate out to a distance of about 1000 km (625 miles),² which is 20% greater than the extent of the entire United Kingdom, though considerably less than that of the United States (New York to Los Angeles is 3,933 km, Miami Fl to Portland Me 2,190 km). See The intermittency statistics of wind power.

It is important to realize that lack of correlation (zero correlation) does not imply negative correlation. Negative correlation between regions is what you would like, as that would mean that if the wind were not blowing in one region, you could be assured that it would be blowing in another.³ In absence of negative correlation, one suspects that any deployment percentage greater than the low-penetration (marginal) capacity factor will result in reduced efficiency (reduced capacity factor and greater cost) and still not keep the lights on without dispatchable backup and/or storage (even greater cost). Statistics alone give no assurance of negative correlation.

Global wind patterns, on the other hand, just might. At least it’s a possibility. At least if the areal extent of the grid greatly exceeds the wind’s positive correlation length. At least in the mid-lattitudes where most of us energy hogs live. In the mid-lattitudes we have prevailing westerlies and trade winds that have served mariners well for at least a millenium, and jet streams too – should they survive global warming in one piece. Which is beginning to look a bit iffy.

Polar jet streams are driven by the temperature difference between the mid-lattitudes and the polar regions. Global warming is not uniform, and the Arctic in particular is warming about twice as fast as the rest. The lower temperature difference results in the northern polar jet stream having slowed from its usual brisk 140 mph clip to a more sedate 100, and has become more meandering (meridonal) in the process.⁴ Arctic warming may nudge the jet stream a bit northward. What that bodes for future wind patterns is for you and me to speculate, and climatologists to try to predict.⁵ One might hazard that past perforance may become an increasingly spotty predictor of future behavior, which may have consequence for planning power grids based upon wind and sun.

Speaking of grids and their 1000 km positive wind correlations, even in the U.S. on wind power alone one would need a fair bit of transmission capacity to shunt wind power over the often long distances between where the wind is blowing and where power is needed. You might not need that transmission often, but when you do, you’ll need it bad. Most of the time that transmission would go unused – see Delivering Stable Electricity. Solar Photovoltaic can help some, but not as much as one might think: fixed-angle PV (e.g. solar-on-roof) generates most power over just a few hours either side of noon for a capacity factor of about 20%, and the sun correlates across half the planet. So intermittent renewables schemes turn to load shifting and demand management. Load shifting is just energy storage: hydro, pumped hydro, batteries, compressed air, thermal. Concentrated Solar Thermal (CST) is one way to store solar energy as heat before it is converted to electricity. Any of these add to the cost of an all-renewables grid. Demand-management just means you and I can’t have all the power we think we need whenever we think we need it. Demand management is proposed to work via smart grids that will detect when such things as smart refrigerators and smart heat pumps and smart EV and PHEV batteries really do need to be powered, and when they really don’t. To a certain extent demand-management is a worthwhile (and probably necessary) pursuit. But smart stuff doesn’t come cheap, and won’t happen overnight.

We’ll visit correlations and costs again in Pandora’s Back Pages: Renewable Economic Models. The bottom line is that the requisite thermal (coal, gas, biomass) backup, and battery and hydro storage, are expected to push deep-penetration intermittent renewables electric costs to over half again – and perhaps twice – what one might reasonably expect for the same power from nuclear. And emit up to ten times the CO2.

2.3 Emissions Density

Table 3 shows the metric tonnes (2,200 lb) of carbon dioxide equivalent emitted when one GWh of electric energy is produced by different power generation technologies. If Table 2 suggests how difficult it is to replace fossil fuels, Table 3 illustrates the necessity:

2.4 Areal Density

The American Nuclear Society estimates “For a 1000-MW plant, site requirements are estimated as follows: nuclear, 1-4 km2; solar or photovoltaic park, 20-50 km2; a wind field, 50-150 km2; and biomass, 4,000-6,000 k${\mathrm{m}}^2$.”⁷

If you don’t think our real estate is valuable, you need a new agent. Site requirements do not include land for mining, so I’m going to skip estimating the total land area required for fossil and nuclear power, and return as time permits. I will say a few words about solar and wind.

As the United States currently generates about 4,000 GWh electricity per year, or 4 PWh, we will roughly estimate the land area required to generate 1PWh from solar power and from wind.
(1 PWh/y) / (8766 h/y) = 114 GW average power.

An excellent reference is Sustainable Energy’s chapters on Solar, and on Wind.

Briefly, the solar irradiance at the earth’s surface is almost an even 1kW per square meter. 1kW is the power incident upon a 1 meter square panel held perpendicular to the sun at noon on a cloudless day, averaged over a year, and not too close to the poles lest the atmosphere pile up. In reality, all days aren’t cloudless, and it isn’t always noon. The “not always noon” part hits fixed-angle solar with a 75% geometry penalty right off the bat. “Into each life a little rain must fall” reduces it still more. One may get effective fixed-angle solar capacity factors by dividing the radiance values from Figure 6.16 on page 46 of the above reference (reproduced as Figure 4 below) by the 1000 W/${\mathrm{m}}^2$ peak.

Wind power density is a bit less. Most planners plan on between 2 and 7 watts per square meter from wind, which Amanda Adams and Harvard Prof. David Keith find a bit optimistic if one actually wants to build out to the 100 GW levels we’re talking here. Adams and Keith⁸ think that one turbine in an operating array might actually reduce the wind energy available to the next, and believe wind power densities on the order of 0.5 to 1 W/${\mathrm{m}}^2$ may be more realistic in a large array. We’ll assume the 1 W/${\mathrm{m}}^2$ value and just multiply all out fixed-angle Solar PV values by 20. If you think wind power density should be greater than that, divide by whatever factor you feel appropriate. But at 1 W/${\mathrm{m}}^2$ that’s 114,000 sq km for 1 PWh/y (Ohio), or 456,000 sq km for the country (two thirds of Texas). As we saw earlier, 1PWh = 114 GWy will require 171,000 2 MW turbines at 33% capacity factor, or four times this – 684,000 turbines to (not) power the country. These would be spaced so that on average each turbine took up 2 sq km (or more). But their towers are only 80 - 100 m high and blades span a bit over a hundred meters, so this would still leave a lot of space for our trivial pursuits.

One would not, of course, site all the country’s wind (or solar) in one place. And 2 sq km per turbine (1.4 km or 7/8 mile inter-turbine spacing) is also an average: one could have relatively narrow strip farms aligned roughly perpendicular to expected wind velocity with much smaller inter-turbine spacing. You just wouldn’t stack them very deep. And some areas – notably the prairie states – just have higher quality wind than others. With our relatively large land area, coasts on two major oceans, and low population density (Table 4), the United States is far better suited than some of our friends to experiment with large-scale intermittent deployments.

A sanity check on the wind values may be obtained from Wind energy: the fastest growing power source:

Which comes out to 175 tonnes per nameplate MW. “The entire wind farm will produce an estimated 650 GWh a year”, so its capacity factor $C_F$ = 650 GWh/y / ( 2 x 80 MW x 8766 h/y) = 46%, much better than onshore 33% (if you’re lucky). 175 tonnes/MW / 0.46 = 380 tonnes/MW capacity factor corrected, compared with 460 tonnes/MW onshore, a considerable savings. 175/0.33 = 530 Tonnes/MW, so the offshore turbines use a fair bit more steel/MW than onshore, but more than make up for it with the better capacity factor. Horns Rev 1 uses steel monopile foundations rather than concrete, which is where the extra steel goes. Engineering tradeoff, one might suppose.

Construction materials for Solar Thermal may be highest of all, but vary by the material used for thermal storage. See NEEDS: Final report on technical data, costs, and life cycle inventories of solar thermal power plants, as cited by Peter Lang in Emission Cuts Realities – Electricity Generation Cost and CO2 emissions projections for different electricity generation options for Australia to 2050. We reproduce some of Lang’s numbers in Table 5 below. Note for nuclear that while steel remains about the same as in Figure 5, the amount of concrete is significantly greater. (More study required to discern cause of the disparity, but Petersen et al. were very thorough.)

Me, I’d just divide by the capacity factors and leave it at that. Yes, wind turbines only last 20 - 25 years. They spin fast and die young. But I suspect come replacement time, their concrete foundations will probably be good for another go or two. As for steel, there your mileage may vary. I don’t know, for example, just how well the steel pylon holds up relative to the nacele and turbine blades. But I doubt come EOL that wind engineers want to replace any more than they have to, and design accordingly.

2.6.1 Wind

We saw in Section 2.4 that 684,000 2 MW turbines might generate the 4 PWh the United States need each year. (Or, they might not...) A 2 MW onshore turbine runs about $3 - $4 million installed,¹¹ so 684,000 of them will only set us back $2 – $2.7 trillion. But we already have 3.6% wind in place, which reduces it to 660,000 turbines at $1.93 - $2.6 trillion. Again, that’s if we can store the windy-day surplus or otherwise even out the intermittency. Else their utility becomes somewhat limited. We might expect the requisite backup and transmission and storage to run us some extra.

2.6.2 Nuclear

If we wanted to power our entire grid with nuclear we’d need 456 GW average, of which 20% is already covered by nuclear, so we’d only need an average 365 GW more, or 365 new 1.1GW plants with capacity factor 90%. At $7.5 billion overnight cost each – what Southern Co. thinks (knock on wood) they are facing for two 1.1GW AP1000 at Vogtle Point – that’s $2.74 trillion. A bit more than the installed cost of wind, but nuclear is dispatchable, and pretty much a drop-in replacement for all the coal and gas we’d want to drop out and replace. We’d probably need to retain some gas as the 365 average GW won’t quite cover all the peaks. Our 8% - 10% hydro might not quite cover them either; the remainder might most effectively be provided by gas. But nuclear can cover peaker loads as well, when it becomes critical.

Another consideration when comparing nuclear to wind is that new wind turbines are expected to last 20 years. New advanced nuclear plants have design life of 60 years: existing plants built in the 1970s had design life of 40 years and usually receive license extensions of an additional 20 years. That might effect how one thinks of capital costs, and 60 year capital amortization would drop levelized cost of new nuclear energy from $108/MWh to $67.

Of course, come replacement time one probably wouldn’t need to replace a wind turbine’s foundation, and possibly not the tower either. As for the nuke, 80 years is pretty far out there. The containment might last longer than that, but it remains to be seen whether early 21st century containment would be suitable for 22nd century needs. Leave that one to the great-grandkids.

We won’t actually deploy that much wind or nuclear “overnight”, either. The cost of each is projected to decrease over time. EIA estimates levelized cost of nuclear electricity will drop from about 11¢/kWh to about 8.5¢/kWh in constant 2009 pennies between 2020 and 2040, wind from 8.5¢ to 7.5¢.¹² These low wind LCOE costs are of questionable utility, and EIA itself cautions against using them. The American Tradition Institute suggests a more realistic Levelized Cost of Usable Energy. This includes cost of fossil backup, which pushes onshore wind cost to 15.1¢/kWh if by gas, and 19.2¢/kWh if by coal, not including carbon capture.¹³

James Conca has run a concise series on energy economics:

The Naked Cost of Energy – Stripping Away Financing and Subsidies

Actual Energy Costs Are Driving Us Towards a Natural Gas Nation

Is the Answer Blowing in the Wind?

The Direct Costs of Energy: Why Solar Will Continue To Lag Hydro And Nukes

Also see Michael Overturf: How The Price For Power Is Set. Levelized costs of UK energy for different technologies are described in DECC: Electricity Generation Costs 2013, pages 17 and 18.

• Jargon and buzz-words can be deceiving. The Nameplate Capacity of a given power source – its nominal peak power output – is meaningful only within the context of its capacity factor, dispatchability, and intended use.
• Capacity Factor is the average power a plant actually produces over an extended period of time, divided by its nameplate capacity.
• Dispatchability indicates whether or not the plant power can be dispatched by the grid operator when he needs it. Subcategories are dispatchable baseload, dispatchable load following, and dispatchable peaking. All thermal and hydro sources are dispatchable, as is CST (concentrated solar thermal). Some are better suited to follow loads or provide peaker capability than others. Baseload-only plants (some older nukes) must be run at nearly constant power, or be shutdown. Restart can take hours or days. Intermittent renewables such as wind and solar PV are non-dispatchable, unless provided with dedicated storage. (Implementation details can vary.)
• Global warming is urgent, but the economics of deep-penetration renewables are not encouraging. Marginal costs at current low grid penetration (0.11% solar, 3.46% wind) do not even begin to tell the whole story, which will be continued in the following article. Be skeptical.

Please remember, your electric utility is in the business of “keeping the lights on.” Your lights on. When you want them on. At a cost you can afford. It isn’t as easy as it sounds.