In recent years, governments around the world have experimented with many different policy tools to encourage the growth of renewable energy. In particular, it is clear that subsidies are needed to stimulate investment in clean technologies like wind and solar that are not yet able to compete effectively on cost alone (especially in the US today, where cheap natural gas is showing the potential to dominate!). Economists, politicians and journalists actively debate the merits and limitations of various subsidies, tax incentives, or of feed-in tariffs in electricity markets, popular in many European countries. However, an interesting alternative is also growing rapidly at the state level in the US: markets for tradable renewable energy certificates (RECs), or as a subcategory, solar renewable energy certificates (SRECs). Here we discuss the vital role that mathematics can play in helping to better understand these important new markets.

Over the last decade or so, about 30 states have implemented specific targets for renewable energy growth as part of a so-called Renewable Portfolio Standard (RPS). Among these, many have a specific “solar carve-out,” a target for the solar sector in particular, in addition to renewables overall. To achieve these goals, about 10 states have launched SREC markets, with New Jersey (NJ) being the largest and most ambitious so far (targeting 4.1% solar electricity by 2028). It is worth noting that similar markets for “green certificates” also exist in various countries around the world.

The basic idea is that the government sets specific requirement levels for solar energy in the state in each future year as a percentage of total electricity generation. Throughout each year, certificates (SRECs) are issued to solar generators for each MWh of solar power that they produce. These can then be sold in the market to utility companies, who must submit the required number of SRECs at each compliance date (once per year). Anyone not meeting the requirement must instead pay a penalty (known as the SACP), which is typically chosen to decrease from year to year, but has been as high as \$700 per MWh in the New Jersey market.

While the concept is straightforward and intuitive (and parallels that of a cap-and-trade market for CO2 emissions), the implementation is far from simple, with different states already trying many variations for setting future requirement and penalty levels. Another important policy consideration is the number of “banking” years permitted, meaning how long SRECs remain valid for compliance after they are first issued (e.g., currently a 5-year lifetime in NJ). A fundamental challenge for regulators is trying to choose appropriate requirement levels many years in advance, such that the market does not suddenly run into a large over- or undersupply of certificates, causing prices to swing wildly.

In New Jersey for example, SREC market prices dropped from over \$600 throughout most of 2011 to under \$100 by late 2012 in the wake of a huge oversupply, and this despite a major rule change passed in 2012 (more than doubling the 2014 requirement) to help support price levels. On the one hand, the large oversupply was good news, signaling the success of the SREC market in enabling solar in NJ to grow very rapidly between 2007 and 2012 (from under 20MW to nearly 1,000MW of installed capacity). On the other hand, this initial success of the market brings with it some risk for its future. At only \$100 an SREC and with the possibility of further price drops, will investors now shy away from new solar projects?

Like all financial markets, SREC markets can provide very rewarding opportunities for investing (in new solar farms in this case). But they also come with significant risk due to volatile price behavior. Financial mathematics, a field that has grown rapidly over several decades now, is well versed in analyzing and modeling such risks and returns. However, most financial mathematicians work on classical markets for stocks or bonds, instead of venturing into the peculiarities of commodity prices, and even more so those of RECs. Nonetheless, commodities, energy and environmental finance is a rapidly growing subfield and popular research area these days (see for example the **May 7th blog post** on the Field Institute’s activities).

So how can mathematical modeling help us to better understand SREC markets? And why is it important to do so? In recent and ongoing work at Princeton University [1], we propose an original approach to modeling SREC prices, which is able to reproduce New Jersey’s historical price dynamics to an encouraging degree. Drawing on some ideas from existing literature in carbon allowance price modeling, we create a flexible framework that can adapt to the many rule changes that have occurred. In particular, we treat SREC prices as combinations of “digital options” on an underlying process for total solar power generation, since SRECs essentially derive their value from the probability of the market being short of certificates and paying a penalty at one or more future compliance dates. However, a key additional challenge comes in capturing an important feedback effect from prices onto the stochastic process for generation. As today’s prices increase, future generation growth rates should also increase (as more solar projects are built), which in turn reduces the probability of future penalty payments, feeding back into today’s price. An equilibrium price emerges, which can be solved for via dynamic programming techniques.

This is an example of a “structural” model, which combines economic fundamentals of supply and demand with tractable stochastic processes and convenient mathematical relationships. Academic literature on energy-price modeling covers a wide range of different approaches and makes use of a diverse set of mathematical tools, from partial differential equations (PDEs) to stochastic processes, optimization and statistical estimation procedures. The feedback discussed above has even been shown to produce interesting applications of complicated “forward-backward SDEs” in the case of carbon markets. Nonetheless, the specific application to SREC markets is extremely new, and we hope to encourage more research in this young and exciting field.

Understanding the behavior of SREC prices is crucial both for investors contemplating a new solar project and for regulators determining how best to design the market or set the rules. How does price volatility vary with regulatory policy? For example, can we effectively implement a requirement growth rule which dynamically adapts to the shortage or surplus of SRECs in the previous year? (as has in fact been attempted in Massachusetts). Can this avoid the need for frequent legislation to rewrite the rules at great uncertainty to all market participants? How can we best avoid sudden price swings, while preserving the attractive features of these markets and their abilities to stimulate growth of solar? While our model allows us to begin to address such important market design issues, many interesting and relevant questions remain to be investigated, and we look forward to continuing to explore this promising new area of applied mathematics!

The reference for our first paper on this topic is given below. For further details on the NJ SREC market, the websites of **NJ Clean Energy**, **SREC trade** and **Flett Exchange** all provide useful and up-to-date information.

[1] Coulon, M.; Khazaei, J.; Powell, W. B.; SMART-SREC: A Stochastic Model of the New Jersey Solar Renewable Energy Certificate Market; working paper, Dept of Operations Research and Financial Engineering, Princeton University.

Michael Coulon

Princeton University

mcoulon@Princeton.EDU