Modeling Cascading Diffusion of New Energy Technologies: Case

Jul 1, 2013 - ABSTRACT: Subsidy programs for new energy technologies are motivated by the experience curve: increased adoption of a technology leads t...
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Modeling Cascading Diffusion of New Energy Technologies: Case Study of Residential Solid Oxide Fuel Cells in the U.S. and Internationally Seth Herron† and Eric Williams‡,* †

School of Sustainable Engineering and the Built Environment, Arizona State University, Tempe, Arizona 85287, United States Golisano Institute of Sustainability, Rochester Institute of Technology, 111 Lomb Memorial Drive, Rochester, New York 14623, United States

‡

S Supporting Information *

ABSTRACT: Subsidy programs for new energy technologies are motivated by the experience curve: increased adoption of a technology leads to learning and economies of scale that lower costs. Geographic differences in fuel prices and climate lead to large variability in the economic performance of energy technologies. The notion of cascading dif fusion is that regions with favorable economic conditions serve as the basis to build scale and reduce costs so that the technology becomes attractive in new regions. We develop a model of cascading diffusion and implement via a case study of residential solid oxide fuel cells (SOFCs) for combined heating and power. We consider diffusion paths within the U.S. and internationally. We construct market willingness-to-pay curves and estimate future manufacturing costs via an experience curve. Combining market and cost results, we find that for rapid cost reductions (learning rate = 25%), a modest public subsidy can make SOFC investment profitable for 20−160 million households. If cost reductions are slow however (learning rate = 15%), residential SOFCs may not become economically competitive. Due to higher energy prices in some countries, international diffusion is more favorable than domestic, mitigating much of the uncertainty in the learning rate.

1. INTRODUCTION Governments subsidize the price of new energy technologies in the hope that learning and economies of scale will reduce costs to make the technology market competitive. Knowledge of future cost reductions and the willingness of consumers to pay for a technology in different submarkets would help plan the level at which technologies are subsidized. Empirically, cost (e.g., $/Watt) and cumulative adoption (e.g., total installed Watts) often follow a power law relationship known as the experience curve.1−4 Consumers in submarkets have different willingness-to-pay, and arranging those submarkets from higher to lower willingness-to-pay value yields a market curve.5 In situations such as Figure 1, an initial subsidy in a technology can stimulate broader competitiveness. We term this cascading dif f usion, because after an initial “uphill” investment, diffusion proceeds from submarket to submarket via market forces. Better understanding of cascading diffusion can support the selection of technologies and appropriate level for subsidy support. Governments face a difficult balancing act between increasing subsidies to promote future price reductions versus lowering them to conserve scarce public resources. Setting a subsidy level that makes a technology attractive in unfavorable submarkets overspends public resources; a subsidy too low even for favorable submarkets will be ineffective. Understanding © 2013 American Chemical Society

Figure 1. Cascading diffusion of new energy technologysubsidy provided until market competitive at point A, then cascading diffusion with no subsidy until no longer competitive at point B.

regional and other differences in willingness-to-pay can help identify a “minimum optimal subsidy” needed to promote a technology. Received: Revised: Accepted: Published: 8097

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was available for Canada, so we included a province with a high population, Ontario.19 We also included countries that might have potential to invest in new technology subsidy schemes for which electricity and natural gas price data was available: Japan, Austria, the Czech Republic, Denmark, Finland, France, Germany, Hungary, Italy, Ireland, the Netherlands, Poland, Portugal, Spain, Sweden, and Switzerland.20 We do not compare the economic and environmental benefits of residential SOFCs with other technologies, for example, Weiss and collaborators analyze learning and diffusion of competing boiler technologies.21 While this is certainly important, the main objective of this paper is to demonstrate the viability of the cascading adoption modeling approach. Future work can refine the method and apply it to a range of technologies.

The notion that technology progresses via adoption in a succession of submarkets has been around for decades. This phenomenon has primarily been discussed in the progression of solar PV technology as additional niche markets opened up.6−9 Tsuchiya5 analyzed prospects for the solar industry in Japan by combining prospective experience curves and market curves based on four types of potential PV consumers (off-grid, commercial, residential, industrial). In this paper we extend the approach of Tsuchiya5 in three directions. First, we develop an approach to construct market curves for different geographic subregions within a region of interest. This is based on the observation that geographical variability in climate and energy prices leads to variability in willingness-to-pay for energy technologies. Second, we compare and contrast national versus international diffusion. Given international spillovers in market scale and learning, considering a broader geographical scope could open up favorable adoption paths. Third, we treat how uncertainty in experience and market curves affects cascading diffusion by scenario analysis of pessimistic and optimistic cases. To demonstrate the method, we analyze a case study of residential solid oxide fuel cells used for combined heating and power in counties that have enough capital to invest in subsidy schemes.

3. MATERIALS AND METHODS The overall method to build the cascading diffusion model is as follows: Develop market curvethe market curve combines willingness-to-pay with potential market size in each submarket to yield a descending stair curve starting from highest and ending with lowest willingness-to-pay. In the case study, submarkets are defined as geographic regions; in general, additional factors can create heterogeneity in markets. Willingness-to-pay is the maximum investment a buyer can make to receive a net economic benefit; in the case study, willingness-to-pay is based on a discounted payback period of five years. The steps in developing a market curve are as follows: • Select set of regions to characterize economic performance of technology, • Model energy flows for the technology’s use in each region, • Develop lower and upper energy price scenarios, • For each region, calculate consumer willingness-to-pay for the technology based on findings from the previous two steps, • Find the number of potential adopters in each region, • Construct lower and upper market curves by ordering regions in descending steps of willingness-to-pay. The step width for each region is the total potential adoption in that region. Build experience curvesassumes future cost reductions follow a power law decrease as a function of cumulative production. The learning rate is an empirical constant that describes the percent decrease in costs for each doubling of production. The steps to develop experience curves are as follows: • Collect data on prices and production level of the technology, • If time series data is limited, find learning rates for related technologies, • Develop optimistic and pessimistic experience curves for future reduction in manufacturing costs. Policy Analysiscombines market and experience curves to estimate the financial sum needed to realize market competitiveness and the degree of total adoption required to achieve market saturation. The policy analysis involves the following steps: • Overlay market and experience curves as in Figure 1 in four combinations: optimistic market, optimistic experi-

2. CASE STUDY: RESIDENTIAL SOFCS FOR COMBINED HEATING AND POWER The residential solid oxide fuel cell (SOFC) is an energy technology currently in development and already subsidized in several countries. The U.S., for instance, provides a $500 subsidy for each 0.5 kW of power capacity for residential fuel cells,10 the U.K. provides feed-in tariffs for microgeneration technologies (including SOFCs),11 and Japan subsidizes 30% of the initial cost and sets a feed-in tariff of around 45 cents/kWh for 10 years.12 SOFCs use natural gas as a fuel input and can be sized to meet a wide range of building electrical loads. When used for combined heating and power (CHP), SOFCs can achieve a nearly 90% first law efficiency.13,14 Due to this high efficiency, an SOFC will use less fuel and carbon compared to other ways of using natural gas (e.g., combined cycle plant, boiler). SOFCs also have low emissions of commonly regulated air pollutants such as SOx and NOx.15 The SOFC used in our case study is a 1 kWe system that runs on natural gas. This size is sufficient to meet the electrical load of a typical single family home for the majority of hours during a year. Any additional electrical needs are provided by the electricity grid. The CHP system is modeled off of the SOFC CHP system presented in Alanne et al,16 who conduct a willingness-to-pay analysis for SOFC CHP systems in Canadian homes. The SOFC is connected to a 1000 L, cylindrical, heat storage tank, filled with water, which handles excess heat from the fuel cell and delivers heat to the space heat distribution and domestic hot water (DHW) systems. The Supporting Information (SI) presents more details of the SOFC system. Developing geographic submarkets assumes consistent differences in economic drivers for different areas. Consistent climate differences between regions are self-evident. The economic performance of SOFCs is also sensitive to electricity and natural gas prices. There are consistent differences in electricity and natural gas prices between U.S. states and other nations (see S4−Sn and S5−Sn in the SI for details).17−20 Given the availability of data aggregated on the state level for electricity and natural gas prices, we chose to conduct our case study of SOFC systems for all 50 U.S. states.17,18 Similar data 8098

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variability in prices since 1990.17,18 The optimistic scenario energy price is set for each state/nation as the year that generated the highest willingness-to-pay, the pessimistic scenario corresponds to the lowest willingness-to-pay. While historical prices are often not a good indicator of the future, it is important to note that variability of willingness-to-pay depends mainly on the difference in natural gas and electricity prices, rather than their absolute values. Because a sizable percentage of electricity is produced from natural gas, their costs are mutually dependent and tend to rise and fall together (with electricity prices generally lagging slightly behind natural gas prices). As natural gas is expected to become more available due to extraction from shale reserves, at least in the U.S. it seems likely that the electricity and natural gas prices will track more closely in the future. Our scenarios likely overestimate the variability in price differences; it is safer to overestimate than underestimate variability. See the SI section “Natural Gas and Electricity Pricing Scenarios” for more information. Energy prices are exogenous in our model. In truth, significant adoption of residential SOFCs would influence natural gas and electricity prices. Modeling the future development and interaction of natural gas and electricity markets, however, is beyond the scope of the current work. 4.3. Willingness-to-Pay. We analyze only the “rational” economic aspects of the adoption potential of SOFCs. That is, we assume adoption occurs if a threshold for discounted payback time is reached, with payback times calculated using discount rates comparable to potential returns on investment. The actual decision of a consumer to invest in an energy technology is more complex, combining direct cash flows with other factors such as perception of the environmental benefits and knowledge of the technology. One approach to account for perception factors stays within a pure economic framework (e.g., net present value) and estimates “implicit” discount rates based on empirical data from past purchase decisions.23 Consumer perception of a technology as environmentally beneficial could reduce the effective discount rate, for example, while reliability concerns could increase it. Another approach is to simulate consumer decisions using an explicitly multicriteria model such as fuzzy logic.24 Such interplay of economic, perception, and knowledge factors is important, but beyond the scope of this work. Willingness-to-pay can be defined as the initial cost that equals the total benefits and costs of a project discounted over time. We choose a discount rate of 10% as one higher than consumers can typically earn on 5−10 year investments, but not one that accounts for indirect factors such as perception that inflate discount rates. We select a 5-year payback time arguing that consumers want to ensure recouping an investment before moving. The median length of time U.S. homeowners stayed in a home was 6 years in 2003−2008, though this is increasing in recent years.25 Residents in Europe and Japan tend to stay in homes longer. Hopefully the 5-year period is a sufficiently conservative measure. For the BAU system, the grid electricity required is simply the residential electrical load as determined by eQuest. The natural gas required is the sum of the natural gas used for the DHW system and the natural gas used by the space-heating furnace. Therefore, the annual fuel cost for operating the BAU system (ABAU) is given by

ence; optimistic−pessimistic; pessimistic−optimistic; pessimistic−pessimistic, • Identify market competiveness crossover points (point A in Figure 1) and integrate area between experience and market curves to estimate minimum subsidy needed to activate cascade, • Identify market saturation points (point B in Figure 1) to estimate total adoption and benefits activated by the subsidy, • Analyze differences between four combinations to suggest policy implications for subsidies. Details of method, data and results for each subanalysis appear in Sections 4−6 of the main text as well as the SI. Note that while willingness-to-pay in a favorable submarket informs the appropriate subsidy level, this does not imply that a subsidy should only be provided to that submarket. For example, if a particular home energy technology is most attractive on average to residents of California, a nationwide (as opposed to statespecific subsidy) still achieves goal of reducing public investment and enables consumers in other regions with different preferences or housing situation to adopt.

4. DEVELOP MARKET CURVES 4.1. Building Energy Modeling. We use eQuest building modeling software to determine the hourly electrical, space heating, and DHW loads for a “typical” single family home over a one-year period. In America, the typical home is estimated to be 2000 ft2 (186 m2), whereas in most of the other nations, the typical home is roughly half the size.22 Interestingly, after conducting a sensitivity analysis, we find that results for willingness-to-pay are insensitive to home size and insulation levels. Details of the sensitivity analysis can be found in the SI. eQuest outputs for a typical home are our business as usual (BAU) case, which uses the electrical grid for household electricity, an electrical air conditioner for cooling, a natural gas furnace for space heating, and a natural gas water heater for DHW. Detailed results can be found in SI Section “eQuest Outputs.” We then use the eQuest generated building loads to model the gas and electricity demands of our SOFC CHP system. The model follows the SOFC CHP system in Alanne et al.16 The 1 kWe SOFC system runs at full power throughout the year. We assume that the SOFC itself has an electrical efficiency of 45% and a thermal efficiency of 45%. Heat generated by the SOFC flows into the heat storage tank and, from there, is distributed to the space heating and DHW systems as required by the building loads. When the energy in the heat storage tank falls below a minimum threshold, the backup furnace is engaged to supply the difference. When the energy in the heat storage tank rises above a maximum threshold, a valve releases the excess heat. Details of the model can be found in SI Section “SOFC System Modeling.” 4.2. Energy Price Scenarios. The economic performance of the SOFC system depends on the energy prices (natural gas, electricity) during the years in which it is operated. In order to account for natural gas and electricity price fluctuations during the operating period of the SOFC system, we establish optimistic and pessimistic energy price scenarios to represent upper and lower bounds of our willingness-to-pay analysis. We estimate the willingness-to-pay for all regions based on historical prices for the five-year period 2005−2009.17−20 We choose this time period because it shows the highest 5-year

ABAU = PelecQ elec,BAU + PNGQ NG,BAU 8099

(1)

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Figure 2. Willingness-to-Pay for a 1 kWe Solid Oxide Fuel Cell (SOFC) in all regions of analysis for optimistic and pessimistic price scenarios. Willingness-to-pay assumes five-year payback period at 10% discount rate.

Where Qelec,BAU is the quantity of electricity consumed annually (kWh) and QNG,BAU is the quantity of natural gas annually required for space heating and hot water (expressed in kWh: 3.6 MJ = 1 kWh). Pelect is the price of electricity in the state/ nation ($/kWh) and PNG is the price of natural gas in the state/ nation ($/kWh). The annual operating cost of the SOFC system (ASOFC) is given by:

The willingness-to-pay for an SOFC system is obtained by discounting future savings compared to BAU system to the present: WTP =

⎛P ⎞ ⎜ , 10%, 5years⎟(ABAU − ASOFC) ⎝A ⎠

(3)

Where (P/A, 10%, 5 years) is the usual formula for the present value of annual cash flows, with a 10% discount rate. We found that willingness-to-pay for SOFC systems is highly sensitive to changes in energy input prices, but weakly sensitive to the characteristics of the houses in which they are placed. Using the eQuest model, all characteristics of the house (size, insulation, etc) become condensed into hourly energy loads. Our SOFC model effectively replaces the first 1 kWh of electric load required by the house with 2.2 kWh of natural gas (45% electrical efficiency). While the SOFC system also stores heat to assist in meeting thermal loads, it is this crucial price swap of 1 kWh of electricity with 2.2 kWh of natural gas that generates the majority of the differentiation between the operating prices of the SOFC system and the business-as-usual case. In this way, the most effective predictor of willingness-to-pay prices for the SOFC system is the difference between the local electricity and natural gas prices. Figure 2 shows the results of the willingness-to-pay analysis for both optimistic and pessimistic energy price scenarios for all regions under study. Not surprisingly, because of high electricity prices relative to natural gas prices, European countries generally have the highest willingness-to-pay. In our optimistic pricing scenario (the one with the largest gap between electricity and natural gas prices) Germany has the

ASOFC = PelecQ elec,SOFC + PNGQ NG,SOFC + mQ elec,SOFConly (2)

Qelec,SOFC is the quantity of electricity purchased from the grid less the quantity sold back via net metering (kWh), QNG,SOFC is the quantity of natural gas used by the SOFC and the backup furnace, m is the annual maintenance cost of the SOFC system per kWh of power generation ($/kWh) and Qelec,SOFC only is the electricity generated by the SOFC. m is assumed to be 0.005 $/kWh.26 We assume the natural gas boiler in the BAU system requires no maintenance over the time period considered. Note that the quantity of grid electricity consumed in the SOFC (Qelec,SOFC) will be less than in the BAU case, and the quantity of natural gas consumed (QNG,SOFC) will be higher. When the SOFC system is run at full power, during some hours of the year it generates more electricity than the electrical load of the residence. During these hours, the excess electricity is sent back to the grid. In our analysis, we take an optimistic outlook toward electricity buyback programs and assume that all states/nations would have retail rate net metering programs by the time SOFCs are adopted. 8100

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highest willingness-to-pay, followed by Denmark, the Netherlands, Italy, and Hungary. Our pessimistic pricing scenario tells a similar story with Denmark having the highest willingness-topay, followed by Germany and Spain. The U.S. states, on the other hand, have much lower willingness-to-pay values, due primarily to access to inexpensive electricity. Predictably, the states with the highest electricity prices have the highest willingness-to-pay for SOFC systems. In the optimistic scenario, Connecticut has the highest willingness-to-pay, followed by Alaska, New York, New Jersey, and California. Leading the pessimistic energy price scenario is Alaska, followed by New York, California, and Connecticut. For some areas, particularly in the U.S., grid electricity is inexpensive relative to natural gas such that willingness-to-pay is negative. Some regions show large differences in willingnessto-pay between optimistic and pessimistic scenarios. Hawaii, which imports almost all of its energy resources, in particular, fluctuates between positive and negative willingness-to-pay. For more information on the importance of the difference in electricity and natural gas prices in determining willingness-topay, see SI section “Natural Gas and Electricity Pricing Scenarios.” 4.4. Market Curves. Willingness-to-pay for each region is combined with the number of potential buyers in each region to generate the market curve. For the U.S., housing data from the U.S. Census Bureau provides the number of single family homes in each state.27 For other nations, we assemble housing data from a variety of sources, including Eurostat, the Japanese Ministry of Foreign Affairs, and Statistics Canada.28−31 The result is the two stair-shaped curves in Figures 3 and 4 for optimistic and pessimistic domestic and international markets.

Figure 4. U.S. market curves with experience curves. State rankings (from highest to lowest willingness-to-pay) for optimistic market curve: CT, AK, NY, CA, NJ, NH, RI, MA, ME, MD, WI, CO, VT, IL, MI, TX, DC, PA, NV, DE, NM, OH, IA, IN, MT, HI, UT, MS, MN, KS, SD, NE, VA. For pessimistic market curve: AK, NY, CT, CA, MA, TX, NH, RI, NJ, VT, NV, WI, CO, ME, MI, MN, DE, IA, NM, IL, MT, UT, WY, MS, SD, OH, OK, PA, NE, IN, KS, ND.

5. DEVELOP EXPERIENCE CURVES The experience curve was first developed to describe cost reductions in aircraft manufacturing1 and is an empirically observed power law decay of some characteristic of an industrial process and cumulative experience implementing that process. In the energy domain, the experience curve takes the following form: ⎛ P ⎞−α C(P) = C0⎜ ⎟ ⎝ P0 ⎠

(4)

where C is the cost of production per energy unit (e.g., $/Wp or $/kWh) at cumulative production level P (e.g., total watt capacity of solar cells produced), C0 and P0 are initial cost and production values, α is a (positive) empirical constant, known as the learning coefficient. α is related to the fractional reduction in costs for every doubling of production, the learning rate (LR), according to α = −log2(1−LR). Equation 4 provides an empirically supported fit for progress for a variety of energy technologies, including photovoltaic, wind, and coal.2−4,32 Developing experience curves for a technology in its initial commercialization phase is a highly uncertain process. In order to bound the possibilities for cost reductions, we establish optimistic and pessimistic experience curves. We refer to an estimate from Neij4 and empirical evidence from Thijssen33 to set the learning rate at 15% in the pessimistic case, and 25% in the optimistic case. Using data gathered from Ceramic Fuel Cells Limited, an Australian company in the early stages of commercial production of residential SOFCs, we construct our experience curve with an initial production volume of 1090 kWe units (P0) and initial cost of $20−37 thousand U.S. dollars (C0) (optimistic and pessimistic values, respectively).34,35 However, only certain components of an SOFC based CHP system are likely to see a reduction in cost as a result of experience. Some costs, such as the heat storage tank, fabrication, and installation are unlikely to be reduced.36

Figure 3. International market curves with experience curves. Country/state rankings (from highest to lowest willingness-to-pay) for optimistic market curve: Germany, Denmark, Italy, Netherlands, Hungary, Spain, Austria, Ireland, UK, CT, AK, Finland, NY, CA, NJ, Poland, Czech Rep., NH, RI, Portugal, MA, ME, MD, WI, CO, VT, IL, MI, TX, DC, PA, NV, DE, NM, France, OH, IA, IN, MT, Switzerland, HI, UT, MS, MN, KS, SD, NE, VA, TN, ND, NC, SC, ID, NY, Japan. For pessimistic market curve: Denmark, Germany, Spain, AK, UK, Finland, Hungary, NY, Austria, Italy, CT, CA, MA, Czech Rep., Netherlands, Portugal, TX, NH, Ireland, RI, NJ, VT, Poland, NV, WI, CO, ME, MI, Japan, MN, ON, France, DE, IA, NM, IL, MT, UT, WY, MS, SD, OH, OK, PA. 8101

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Table 1. International Case: Fuel Cell Diffusion Paths and Minimum Subsidies Needed for Combinations of Optimistic and Pessimistic Experience and Market Curves scenario

minimal subsidy path

average subsidy

optimistic experience, optimistic market optimistic experience curve, pessimistic market optimistic market, pessimistic experience curve pessimistic market, pessimistic experience curve

$53 million initial subsidy in Germany to 20 000 units makes market competitive, after which cascade flows in Germany and other nations to around 160 million units $140 million total subsidies in Denmark to 70 000 units to make market competitive, after which cascade flows to 100 million units over various countries $5 billion initial subsidy in Germany to 3 million units to make market competitive, after which cascade proceeds in various countries to 70 million units SOFCs never become economically attractive for residences, but $240 billion total subsidies will make competitive in 100 million units

33 cents per device for 160 million units $1 per device for 100 million units $72 per device for 70 million units $2600 per device for 100 million units

Table 2. U.S. Case: Fuel Cell Diffusion Paths and Minimum Subsidies Needed for Combinations of Optimistic and Pessimistic Experience and Market Curves scenario optimistic experience curve, optimistic market optimistic experience curve, pessimistic market optimistic market, pessimistic experience curve pessimistic market, pessimistic experience curve

minimal subsidy path

average subsidy

$170 million initial subsidy to 97 000 units makes market competitive, after which cascade flows to around 50 million households $430 million initial subsidy to 390 000 units makes market competitive, after which cascade flows to around 22 million households SOFCs never become economically attractive for residences, but $190 billion total subsidy will make competitive in 50 million homes SOFCs never become economically attractive for residences, but $240 billion total subsidies will make competitive in 50 million homes

$2 per device for 50 million units $20 per device for 22 million units $3800 per device for 50 million units $4700 per device for 50 million units

From Braun,37 we found estimates of the cost of these elements to be ∼$800. As a result, we modified our cost equation to include a floor price for the SOFC system: CSOFC(P) = 800 + (C0 − 800)P log 2(1 − LR)

As can be seen in Figure 4, WTP(P) is a descending staircase function for successive willingness-to-pay in different regions. The result of the integral is:

(5)

min. subsidy(Pt) =

Results for optimistic (in red) and pessimistic (in blue) experience curves are shown in Figures 3 and 4.

n−1

+

∑ WTPj#householdsj j=1

6. POLICY ANALYSIS

+ WTPn(Pt − Pn)

Figures 3 and 4, inspired by Tsuchiya,5 overlay the international willingness-to-pay results as market curves with optimistic and pessimistic experience curves. The two experience curves provide the uncertainty bands for the price projections of this technology, while the two market curves provide uncertainty bands for its potential demand. Visual inspection of Figures 3 and 4 leads to two central conclusions. First, the value of the learning rate is critical to realize willingness-to-pay that exceeds production costs. For learning of 25%, costs soon fall below willingness-to-pay, for 15% learning, cost always exceeds pessimistic willingness-to-pay. Second, international versus domestic perspectives yield very different potentials for residential SOFCs, and presumably, other energy technologies. Compared to the U.S., higher willingness-to-pay in Germany, Denmark, and a few other European countries has the potential to more easily reduce costs to make the technology market competitive. The results in Figures 3 and 4 enable quantitative analysis of the amount of public subsidy needed to make residential SOFCs competitive in various markets. In general, when cost > willingness-to-pay, the minimum subsidy needed to bring the cost from initial value C0 to some target Ct(Pt) is given by integrating the difference between the curves: minimum subsidy(Pt) =

1−α ⎡ ⎤ C0P0 ⎢⎛ Pt ⎞ − 1⎥ ⎜ ⎟ ⎥⎦ 1 − α ⎢⎣⎝ P0 ⎠

∫P

Pt

0

(7)

Where n is the number of steps in WTP(Pt) for a given value of P, WTPj is the value in region j, #households is the number of detached homes in region j, and Pn is the production value for the left side of the nth step. The minimum subsidy assumes sequential adoption in favorable regions, being fully adopted in one region before moving to the next region with lower willingness-to-pay. Clearly this is an abstraction of actual diffusion, but the optimal subsidy path still provides valuable information for policy planning, in particular for identifying initial adoption needed to make the technology competitive in a larger set of markets. The minimum subsidy depends on the specific experience and market curves. We analyze the four combinations of optimistic and pessimistic curves in Figures 3 and 4. Two patterns emerge: For three cases (optimistic experience− optimistic market, optimistic experience−pessimistic market, pessimistic experience−optimistic market), the pattern is an initial phase of subsidy to bring cost down to willingness-to-pay, after which adoption is driven by market forces until a point when willingness-to-pay falls below cost again. We assume that public subsidy would only be used for the initial phase to match cost to willingness-to-pay, then withdrawn until market-driven adoption ends. Using eq 7, we calculate the cumulative subsidy needed to initially bring cost down to willingness-to-pay. The second pattern is for the pessimistic experience−pessimistic market combination: Cost never falls below willingness-to-pay. The two curves fall in tandem until adoption in around 90 million homes, after which willingness-to-pay falls much more

⎛ ⎛ P ⎞−α ⎞ dP ⎜⎜C0⎜ ⎟ − WTP(Pt)⎟⎟ ⎝ ⎝ P0 ⎠ ⎠ (6) 8102

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(e.g., state, nation), though in fact there is a distribution of submarkets within each region. Variability in willingness-to-pay within regions could qualitatively change the market curve. The perceptions of consumers of the technology could shift the market curve or even call for integrating the model in a larger multicriteria decision simulation (e.g., Zhai and Williams 2012).24 Also, the current model considers only one technology. The decision to purchase one technology is influenced by the presence of other exclusive and nonexclusive options. Mass adoption of SOFCs could also influence natural gas and electricity prices. The results raise the issue of potential benefits from interregional cooperation on energy technology subsidies. Current subsidy policy is, by and large, a domestic affair, done within states or nations. Cooperation complicates policy making, but depending on the technology, could deliver benefits justifying the difficulty. Note that development of manufacturing to produce new energy technologies is a benefit as well. If country A produces a technology and sells with the support of a subsidy in country B, both benefit. It is reasonable that both parties contribute to the subsidy. When and how such cooperation should take place remains an open question.

rapidly than cost. In this case we analyze subsidy required until adoption at 90 million. The average subsidy per device is obtained by dividing by the total subsidy to activate the market (point A in Figure 1) by the total number of units and market driven adoption (Point B in Figure 1). Results of analysis for the international case are shown in Table 1. An optimistic experience curve yields promising adoption paths: Relatively small initial investments in high willingness-topay countries such as Germany, Denmark, and/or Spain could make the technology market competitive in many households. The pessimistic experience curve is not promising for diffusion. With the pessimistic market curve, substantial subsidy is needed throughout diffusion. Unless resiliency and/or environmental benefits in addition to carbon are (very) highly valued, residential SOFCs do not look attractive. Table 2 shows results of a similar analysis considering only the United States. Because willingness-to-pay is lower in the U.S. compared to some European countries, the subsidy needed to make SOFCs competitive is higher in purely domestic diffusion compared to international. With a pessimistic experience curve and optimistic market, in the U.S. only, SOFCs never become economically attractive.

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7. DISCUSSION SOFC Case Study. Within the scope of factors considered, the attractiveness of residential SOFCs for public subsidy depends critically on the learning rate. If the learning rate is high (i.e., 25%), relatively small initial subsidies can potentially activate cascading diffusion leading to substantial public benefits. While an international cascade lowers subsidy requirements, there are many nations (e.g., U.S., Germany) that could justify a SOFC subsidy program from a purely domestic perspective. For lower learning rates the situation is more complicated. Lower learning leads to risk that subsidies will not bring the technology to market competitiveness and therefore not be an effective use of public funds. An international cascade reduces the risk, allowing for favorable cascades for lower learning rates. Two policy actions could manage the uncertainty in learning rates. One is detailed study of learning and scale cost reductions for residential SOFCs. A second is a provisional subsidy of residential SOFCs to determine the experience curve with actual “experience”. Unless consumer perceptions of the benefits go beyond its direct economic implications, the subsidy needs to be large enough to make the technology market competitive (it is currently not in any nation). After a probationary period, the learning rate can be determined to decide the future of the subsidy. As caveats, we reiterate that the results are contingent on data quality, modeling, and the assumption that a five year discounted payback time (w/discount rate = 10%) is a reasonable measure of economic attractiveness to homeowners. Also, the purchase decision by homeowners is driven not only by purely economic factors; behavioral and knowledge issues can also play significant roles. The Cascading Diffusion Approach. The method is widely applicable to a variety of energy technologies. While the model simplifies a more complicated system, we argue that this approach is close to the simplest data-driven model that captures important qualitative behaviors. This said, if numerical results are to be applied in policymaking, a number of issues should be explored, including, but not limited to the following: Markets are treated here as uniform within a geographic confine

ASSOCIATED CONTENT

* Supporting Information S

Included in the Supporting Information document are (1) the input selections for eQuest, (2) sample building load outputs from eQuest, (3) a detailed description of the SOFC system model, (4) sample willingness-to-pay calculations, (5) a detailed explanation of the fuel input price scenarios, (6) calculations of estimated CO2 savings from SOFC systems, (7) sensitivity analysis and uncertainty. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: 585-475-7211; e-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research is supported by two grants from the National Science Foundation: CBET-093383 from the Environmental Sustainability program and EFRI-0836046 in Resilient and Sustainable Infrastructures. We thank the reviewers for extremely valuable feedback, their comments led to significant improvements in the analysis.

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REFERENCES

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Policy Analysis

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