Balancing Math and Story Time

Last week’s edition of The Economist published a story about how the U.S. Federal Reserve System writes the “Beige Book,” which is a short collection of anecdotes about the experiences of specific (but anonymous) people and businesses on things like wages, hiring, orders, etc. (“Storytime with the Fed: Low inflation means the Federal Reserve is changing whom it listens to” July 20, 2019) The Fed. compares these stories against their various analyses to form a better picture of what is going on in the economy. Apparently, the Beige Book really does affect their view points. The Economist’s article somewhat disdains their use of stories rather than just sticking with the numbers. After all, shouldn’t serious people follow the facts rather than getting emotional about heart-tugging stories?

My experience has been that indeed stories, paired with good analysis, are essential to putting together a clear picture of something. Without the stories, one can make the wrong conclusions from the analysis or worse, can ask the wrong questions entirely. People can also make mistakes with analysis, especially if it requires complicated math (or even simple math). Stories can be the gut check for this. However, without the analysis, viewpoints can only cover the breadth of available examples. If the only lesson learned you have is a story about something, you are only going to be able to understand cases similar to that story.

In one project several years ago, I was working in a team designing a ceramic-tube heat exchanger. Getting meaningful samples to play with took a long time since we were working with specialty stuff. In the meantime, we continued along with simulation work, spreadsheet calculations, and CAD, accounting for things like thermal expansion, material strength, and the probabilistic nature of crack development. Finally, I got an elegant sample of a tube with a tube joint and immediately started playing with it on my desk upon receiving it. To my horror, I knocked it over and broke it into pieces just hours after taking it out of the box. At that point, it finally sank in that what I was working with was more fragile than a nice wine glass. Sure, the design satisfied all of the load cases, but how was anyone going to handle these things? Our analysis was missing key practical questions until we started developing experiences with samples, in other words by developing anecdotes.

Another case comes from Michael Lewis’ “The Big Short” (Okay, I only saw the movie on this one.). In one fun scene, a hedge fund team in New York is trying to understand whether there is a bubble in the housing market. The data associated with mortgage-backed securities are so complicated that it is hard to make sense of it. They finally confirm their suspicion of a bubble by heading down to Florida to talk to people. A brief trip of talking to a real estate agent, mortgage brokers, and a stripper who owns six homes illustrates to them that mortgage practices in the region are highly unethical and are only temporarily sustained by quickly rising home prices.

On the other hand, relying only on anecdotes can mean taking the scenic route to success. Once while working with Richmond High School’s FIRST robotics team (Team 841 !), a group of students spent 3 or 4 hours and built a quick-and-dirty wooden mock-up of a ball shooter designed to hit a target 10 feet up and 6 feet forward. They used available motors, wheels, and other parts from the last ball shooter they had built the previous year, and they made something like a baseball pitching machine sized for a large Wiffle ball. On the first try, the ball dribbled out of the shooter and went about a foot in the air. This underwhelming artillery demonstration succeeded in providing motivation to work out some math and physics. This showed that their driving wheel needed to be spinning about 8 times faster. After some analysis, the ball shooter was sized appropriately and began to nail the target.

Trying out a mock-up of a ball shooter with FIRST Robotics Team 841 at Richmond High School. And yes, they do wear their safety glasses.

It may sometimes be hard to jump back and forth between analysis and story time, but the results turn out better doing both. The Fed. has even formalized anecdotes into part of their process, and they are better off for it. So, if that computer has been getting too much attention, it might be time to go talk to people. For those extroverts out there, it may instead be time to sit quietly and work out some math. Either way, do both.

Why haven’t we gotten green hydrogen for the Hydrogen Economy?

In his 2003 State of the Union speech, President Bush declared about hydrogen-fueled cars:

“With a new national commitment, our scientists and engineers will overcome obstacles to taking these cars from laboratory to showroom, so that the first car driven by a child born today could be powered by hydrogen, and pollution-free.”

Back in the early 2000s, the Hydrogen Economy had become all the rage in energy and environment discussions, and even the president caught the bug. The hydrogen car was going to be awesome because it was going to use a fuel cell that made emissions consisting of only water. While making a commercially-viable hydrogen fuel cell car a reality would be a major feat of engineering, government policy, and business, the real linchpin was where the hydrogen was going to come from. Getting it from natural gas would be a bummer because the whole point was that it should be a clean fuel! It needed to come from solar or wind power.

Today, people born in early 2003 may now be motorists, and there are indeed a few hydrogen fuel cell vehicles on the road, proving the technical feasibility. The solar industry is also big and growing, and wind power has definitely gone mainstream. These solar and wind projects nearly all make electricity though and not hydrogen for fuel cell vehicles. If we have the renewable energy and the fuel cell cars, why hasn’t renewable hydrogen fuel come to fruition at scale?

Two factors (among others) act to divert attention away from building capital projects to make renewable hydrogen. First, a solar plant (or wind or hydroelectric) is more competitive at generating electricity than making hydrogen. Second, if developers did want to build a solar hydrogen plant, they would likely find other applications more attractive than providing hydrogen fuel for cars. Neither of these are reasons why solar hydrogen for cars is a bad idea but just that this may not be the best option for those considering spending money on a renewable energy project. Finally, those set on delivering renewable hydrogen for cars may find that building a dedicated plant is not the easiest way to go.

How Hydrogen Is Made

First, let’s digress on how hydrogen is made. Three general methods exist at large scale:

  1. Electrolysis. In electrolysis, electricity is used to split water molecules into oxygen and hydrogen with an energy efficiency of approximately 70% to 75% for a large plant. (Santos 2012)
  2. Steam Methane Reforming (SMR). (or steam naphtha reforming of the light fraction of crude oil) Methane is reacted with water to make carbon monoxide and hydrogen. A second reaction between carbon monoxide and water produces more hydrogen. A large SMR hydrogen plant that is integrated into a chemical plant, like an oil refinery can achieve an efficiency of 80% to 85% on a higher heating value basis (process plant analysis).
  3. Gasification of coal (or biomass). Coal is reacted with water and a limited amount of oxygen to make carbon monoxide and hydrogen (used as “town gas” in the late 19th & early 20th centuries). A water-gas shift reaction is used to make more hydrogen from carbon monoxide and water.

The large majority of hydrogen produced in the world is from natural gas and from oil.

Why Not to Make Renewable Hydrogen

Of the three methods above, renewable hydrogen can be made at large scale by making electricity with a solar, wind, or hydroelectric plant and then using an electrolysis plant to make hydrogen. A rare example plant built in the 1980s is a 165MW-sized electrolysis plant located at the Aswan Dam in Egypt. Hydroelectric power is used to make hydrogen, which is used to make ammonia for fertilizer. 

A good reason why lots of utility-scale solar electricity plants are being built but not solar hydrogen plants (or wind-hydrogen plants) is because solar is more competitive against natural gas for making electricity than for making hydrogen. Comparing efficiencies illustrates this. For solar, the efficiency to generate electricity is significantly higher than to make hydrogen. For natural gas, the efficiency to generate electricity is significantly lower than to make hydrogen. A solar electricity plant may be 16% efficient (sun to AC power under good conditions), and an electrolyzer can be 70% efficient. The overall sun-to-hydrogen energy efficiency would be 11% (0.16 * 0.7 = 0.11) versus 16% to only make electricity. A combined cycle natural gas plant can deliver power with a 50% efficiency (HHV); whereas, hydrogen can be made from natural gas with an efficiency of, say 80% in an SMR plant integrated in a large chemical plant complex. This comparison does not show that a solar hydrogen plant should not be built. It simply shows that a project developer will probably make a stronger economic case using the solar field to generate electricity.

  Making AC Power Making Hydrogen
Solar 16% Efficiency 11% Efficiency
Natural Gas 50% Efficiency 80% Efficiency

Suppose you did build a solar hydrogen plant

If a group of developers had a great plan to build a solar hydrogen plant to deliver low-cost hydrogen, they would likely find easier customers than a small, emerging fleet of fuel cell cars that needs a distrusted network of refueling stations. The bigger issue is that there are already major markets for hydrogen, totaling over $100 Bil. globally, and these existing applications are much easier to serve. First, oil refineries use a substantial amount of hydrogen to hydrocrack large hydrocarbons molecules into smaller ones and also to drive the sulfur out of refined products (hydrodesulfurization!). Second, hydrogen is a building block to make ammonia which is used in nitrogen fertilizer. In both cases, a substantial amount of hydrogen is used at a single process plant. Making these applications still easier to serve in two locations, the U.S. Gulf Coast and northern Europe, is that hundreds of miles of hydrogen pipelines have been built to move the stuff between producers and industrial consumers.

How can you get renewable hydrogen for fuel cell cars then? (not just trucking hydrogen from a steam-methane reformer plant)

Hydrogen refueling stations for cars may not attract investment for a large, dedicated plant. In a growing number of places, a business can contract for renewable electricity and use an electrolyzer to make the hydrogen. For example, here in the northern San Francisco Bay Area MCE (formerly Marin Clean Energy) provides different percentages of “green” electricity to its customers by contract and procures the power from wind farms, solar plants, and hydroelectric facilities. This may not look like a renewable fuels plant, but this method can enable small-scale hydrogen production with very low greenhouse gas footprint. That hydrogen from electrolysis is significantly more expensive than hydrogen from steam methane reforming serves as yet another hurdle toward this technology’s adoption. Though not impossible to deliver renewable hydrogen, it is no wonder investment money is flowing toward electricity plants or other applications.

Getting to a Fleet of Solar Cars

I told my brother-in-law that I had previously worked on car engines but now work at a solar company. He then said, “Oh really? Can you make a solar-powered rocket car? I want that.” (Sure, buddy…) Is he really in the market for the sustainable Batmobile? Like the majority of Americans, he buys used cars that burn gasoline. He’d be happy to use renewable fuel if it was cost-effective and available. If my brother-in-law is willing to compromise on the rocket engine, maybe we can still figure out how to do solar-powered cars.

The 1966 Batmobile….Powered by Solar Hydrogen?

There are a number of ways to power a car with solar energy. Putting solar panels on the roof of a car unfortunately won’t generate enough power so generating power with a stationary plant and using storage is required (Sono Motors would disagree). Generating electricity with a solar plant and then using the power to drive an electric car is one way. Biofuels, like ethanol, might be considered solar fuels, except that a substantial amount of fossil energy is typically used to produce them, so the energy is only partly solar energy. A solar plant could be used to make hydrogen by driving an electrolysis plant, and the hydrogen could be used in a fuel cell vehicle like Honda’s Clarity Fuel Cell Vehicle (FCV). Finally, solar plants could be used to make hydrogen and carbon monoxide feedstocks to make synthetic gasoline or diesel fuels, which could be used in a normal car. Oh yeah, a solar plants could also be used to make hydrogen, which could be liquefied and stored in a tank on a car and then burned in a rocket engine. (Rocket cars do pose other challenges like projecting hot exhaust and driving in reverse.)

Rather than talking about one car, more interesting questions are how to power a large number of cars, say half of the vehicle fleet, with solar energy and how to get there quickly. What is the easiest strategy to get solar-powered vehicles to become widely used?

The first point about automotive power trains is that the fuel and the engine need to work well together. Indeed, gasoline and the spark-ignition engine have evolved together over the last hundred years. If a solar fuel is going to work, it needs to work with a solar fuel engine. In addition to large solar fields converting sunlight, one of the following scenarios is needed:

A) A new engine that works with already available fuel infrastructure,

B) A new engine and a new fuel with all of the required infrastructure, or

C) A new fuel with new infrastructure that works well with existing engines.

Electric vehicles are an excellent example of a “new engine” that works with an existing fuel: electricity (Strategy A). Solar plants or any other power plant can feed electricity to the grid, and inexpensive charging stations can be used to recharge electric cars. The main issue with this strategy for making solar-powered cars ubiquitous is that it is slow, simply as a result of the sheer scale of the vehicle fleet. There are about 270 million passenger vehicles in the U.S., and only about 17 million new cars were sold in 2018. So, roughly 6% of the fleet gets replaced each year. In 2018, 360,000 electric vehicles were sold in the U.S. This is about 2% of total new car sales, and that represents replacing just 0.1% of the total vehicle fleet. Besides convincing people to buy new-technology cars, people just don’t buy new cars that often.

If getting a new engine car into the fleet is slow, pursuing a strategy of using a new engine and a corresponding new fuel is even slower and harder (Strategy B). A concept like producing hydrogen with a solar plant and then running cars off hydrogen in a fuel cell would require 1) the vast adoption of new, expensive vehicles and also 2) the capital investment of duplicating all of those gas station, tanker trucks, pipelines, etc. with hydrogen infrastructure. The green car market is getting competitive too. Any new fuel cell vehicle market entrants will have to convince green-minded shoppers not to buy one of the many available electric car models that can be charged at many charging stations and probably also at the owner’s house. It may be a winner-take-all scenario for the green car market between fuel cell vehicles, electric vehicles, and other green cars.

The best strategy is to introduce a solar fuel that utilizes both the existing vehicle fleet and the existing fuel infrastructure. That way all you have to do is make the fuel. Although this is no mean feat, it is only one third of the system of fuel production, fuel distribution, and fuel consumption. More importantly, it is much faster because there is no delay waiting for new cars to trickle into the vehicle fleet. What this looks like is a large-scale solar plant that generates some kind of feedstock that is converted into a drop-in replacement for gasoline or, for the heavy truck sector, diesel fuel. The first stage of introduction would be a feedstock that is blended into one of the process streams at an oil refinery to back out some amount of crude oil or perhaps contributing solar hydrogen to replace a refinery’s hydrogen production.

An example to follow is the biofuels industry. U.S. oil imports were growing quite high in the 1990s so the government incentivized biofuel production and consumption. During the 2000s, ethanol consumption in the transportation sector consequently shot up by a factor of 10 to around 4% of the total transportation energy consumption. A normal engine cannot operate properly on straight ethanol so most ethanol was blended into gasoline at fraction of up to 10%, which a normal spark ignition engine can accept. Blending is done far from the consumer, and the average motorist does not have much choice but to use the gasoline with ethanol in it. The introduction of ethanol has been fast and effortless on the part of the consumer. A solar feedstock might follow a similar path. 

The sexiest way to do a solar car would be a solar-hydrogen-fueled rocket car. However, this would be just one, expensive car. The fast way to get to a fleet of solar cars may be through the oil refinery. Detailing how this can start, and why we haven’t seen activity in this space comes next.

Anacortes Refinery, Photo credit: Walter Siegmund.

Applying Experience Curves to Utility-Scale Solar Electricity

My father worked for IBM in the 1960s, and one time someone showed him a megabyte of memory. The machine filled a whole room. It reportedly cost $1MM, and it was a big deal. Checking on Amazon, you can buy a 64GB thumb drive for $13. Unpacking this, we can adjust $1MM in, say, 1965 to $8MM in 2018 dollars using the consumer price index. So, in 2018 dollars, this early megabyte of memory apparently cost $8MM/GB. Today, it costs $0.20/GB, which is an 11 order-of-magnitude difference! Could they have predicted in the 1960s that memory would become ubiquitous and near free?

What about solar power? According to LBNL’s 2018 Utility Scale Solar report, solar energy now makes up about 2% of the electricity used in the U.S.1 It is beginning to achieve market penetration but is still frequently grouped into the “Other” category in power generation pie charts. Will costs of solar power fall substantially, and will solar power become ubiquitous? There may be reasons besides cost that holds back the solar industry, but lowering cost always helps to expand a market.

There are good cost data since about 2010 for utility-scale solar plants, defined by LBNL as plants larger than 5MW, so cost trends can be developed. LBNL’s data, replotted in Figure 1, indeed show a precipitous decline in the price of solar power as the total solar capacity increases.

Figure 1. Average Power Purchase Agreement Price (left axis) vs. Vintage Year of Contract. Also, Cumulative Capacity (right axis) vs. Year.

One way to look at cost trends is with an experience curve, sometimes called a learning curve. Developed by BCG in the 1960s, the idea is that there may be an empirical correlation between the cost of producing something and the total amount of that thing that has been produced.2 As a group or an industry makes more of something, it somehow finds ways to reduce cost. A power law curve fit can be applied, and one might talk about a reduction in cost, say 20%, for every doubling of total goods produced.3 So, as shown below, Cn is the cost of the nth unit produced. C1 is the hypothetical cost of a first unit (really just a curve fit parameter). n is the total number of units produced. Finding the exponent a can be used to show the cost reduction percentage for each doubling of the number produced.



% Reduction for Each Doubling = 1 – b

Back to LBNL’s solar data, the price of solar versus total solar capacity (utility-scale plants only) can be fit reasonably well to a power law curve, as in Figure 2. This curve is saying that for each doubling of capacity, the PPA price fell by roughly 20% (And, you can tell your nerd friends you saw log base 2 used for something.).

Figure 2. Solar Power Price (PPA) vs. Cumulative Capacity.

The trick now is to learn something from this. This empirical curve is saying that the U.S. solar industry is getting better and better at offering low-cost solar power as it gets more experience and, without saying why, that the rate of improvement follows a trend. What this curve does not show are the conditions required for the curve to continue.

The rate of cost reduction could slow or stop if various factors changed. First, if utility customers stopped asking for lower prices, power plant developers would happily stop lowering them. This might happen because customers instead asked for performance. Adding batteries to solar plants would deliver more value at higher cost, for example. The government can affect things. For example, the Federal Investment Tax Credit is set to step down and expire, and this may result in a temporary pause in cost reduction. Increased regulations could also halt cost reduction. Another implicit assumption in this plot is that PPA price is shown, and solar power does not have to be sold in a power purchase agreement paradigm. In this system, an agreement is made before the plant is build that the off-taker agrees to buy power for many years at a set price. Solar plants could be built with no contract price and could instead sell power at market prices. This probably would result in higher average solar power prices since it forces the seller to assume more risk.

Assume though that confounding factors do not ruin the trend. With a 10X increase in deployed solar capacity, this curve extrapolates the PPA price to under $15/MWh, as in Figure 3.

Figure 3. PPA Price Extrapolated vs. Cumulative Utility-Scale Solar Plant Capacity. The price is extrapolated out to a 10X increase in total capacity using the experience curve.

One might wonder that it will take a long time for an additional 350 GW of solar plants to be built in the U.S. when only 6 GW were built in 2017. (The math comes out to 58 years).

One factor that may increase demand for solar power plants is the relative cost compared with other options. The “team to beat” in power generation these days is the combined cycle natural gas plant, fueled by low-cost shale gas. The price of natural gas has bounced around $3/MMBtu for a while.4 Combined cycle power plants perhaps do 50% efficiency on a higher heating value basis. So, the fuel expense for natural gas electricity comes out to $20.50/MWh. So, the extrapolated solar power PPA price with a 10X scale-up of the solar plant fleet is 25% lower than the current fuel OPEX for efficient natural gas plants. This comparison should be motivating for utilities if these numbers in fact come to pass.

Of course, this extrapolated conclusion is riddled with assumptions. It costs more than just fuel to run a natural gas plant. The cost of natural gas is not static. There is a wide range of costs for solar plants, not just a single number. Lots of things can change the cost reduction curve for solar. Nevertheless, there is a general trend.

Will solar power become ubiquitous and cheap? It seems to be on that path for the foreseeable future.



  1. Bollinger, M. and Seel, J., “Utility-Scale Solar: Empirical Trends in Project Technology, Cost, Performance, and PPA Pricing in the United States – 2018 Edition, Lawrence Berkeley National Laboratory, September, 2018. https://emp.lbl.gov/utility-scale-solar
  2. “The experience curve,” The Economist, Sep. 14, 2009, https://www.economist.com/news/2009/09/14/the-experience-curve
  3. “Experience Curve Effects,” Wikipedia, https://en.wikipedia.org/wiki/Experience_curve_effects
  4. “Natural Gas: U.S. Average Natural Gas Price,” NASDAQ, March 2, 2019, https://www.nasdaq.com/markets/natural-gas.aspx?timeframe=2y

A Book Review: _Competing Against Luck_ by Christensen, Hall, Dillon, and Duncan

This is the post excerpt.

Competing Against Luck: The Story of Innovation and Customer Choice by Clayton Christensen, Taddy Hall, Karen Dillon, and David Duncan, 2016.

Christensen and his team start with a McKinsey poll that 84% of global executives acknowledge that innovation is extremely important to their growth strategies but 94% were unsatisfied with their own innovation performance. In 2015, U.S. public companies spent $680 billion on R&D showing innovation is indeed important. Since perceived need and effort are not the problems, Christensen and team conclude innovation efforts are directed in a haphazard way and propose a “jobs to be done” theory for products. What job is a customer hiring a product to do? People want a quarter inch hole and so they go to the hardware store to buy a quarter inch drill bit. No one actually cares about drill bits. The book discusses how you figure out exactly what it is your customer needs and how to provide that for them. Why does a company ask what someone’s favorite milk shake flavor is when the most important thing to them that it fits in the car’s cup holder? Why would V-8 be marketed as a good tasting drink when people buy it to replace cooking vegetables? The book teaches how to ask the right questions when marketing products.

As an engineer, it’s heart-breaking to spend blood, sweat, and tears to get something working only to find out that no one wants it. This well-written book provides hope that misdirected development is avoidable and shows how to ask the right questions to tune what you are doing to what someone wants. Certainly, others have written books on this topic, and the authors acknowledge as much. But for those who haven’t already got a favorite on the topic, I thought this book was great and would recommend it.