Evaluation and Optimization of Feed-In TariffsI Kyoung-Kuk Kima , Chi-Guhn Leeb,∗ a Industrial b Mechanical

and Systems Engineering, KAIST, South Korea and Industrial Engineering, University of Toronto

Abstract Feed-in Tariff Program is an incentive plan that provides investors with a set payment for electricity generated from renewable energy sources that is fed into the power grid. As of today, FIT is being used by over 75 jurisdictions around the world and offers a number of design options to achieve policy goals. The objective of this paper is to propose a quantitative model, by which a specific FIT program can be evaluated and hence optimized. We focus on payoff structure, which has a direct impact on the net present value of the investment, and other parameters relevant to investor reaction and electricity prices. We combine cost modeling, option valuation, and consumer choice so as to simulate the performance of a FIT program of interest in various scenarios. The model is used to define an optimization problem from a policy maker’s perspective, who wants to increase the contribution of renewable energy to the overall energy supply, while keeping the total burden on ratepayers under control. Numerical studies shed light on the interactions among design options, program parameters, and the performance of a FIT program. Keywords: Feed-in tariff, Renewable energy, Real option

1. Introduction Feed-in tariffs are a generic class of policies that guarantee a set payment, a “tariff”, for electricity produced from pre-specified sources that is “fed” into the power grid. This provides investors, including small-scale developers like homeowners as well as medium to large-scale companies, with incentives to participate in such programs by securing certain returns on their investments. As of today, it has been adopted by more than 75 jurisdictions and many of these governments have experimented with a variety of program options. The Public Utility Regulatory Policies Act in the National Energy Act of the US in 1978 is regarded as the first FIT policy (Couture et al. [5]). The implementation of this law was left to states and California pioneered in the FIT program by establishing Standard Offer Contract No. 4, which set the payment based on the estimated long-term avoided cost of conventional generation. Standard Offer No. 4 was the first successful FIT but no new offer contracts have been made since the collapse in the price of oil in 1984. In Europe, FIT programs have been believed to be the primary contributor to the success of the renewable energy markets in Germany and Spain (Couture et al. [4]). In 1990, Germany adopted “Stromeinspeisungsgesetz”(StrEG), a law enforcing utility companies to purchase electricity from renewable energy generators at prices tied to the retail price of electricity, unlike Standard Offer No. 4 setting the rate proportional to the avoided wholesale cost. I The authors would like to thank Thomas K.-C. Fung for his excellent research assistance. The authors also thank the anonymous referees for detailed comments which greatly improved the manuscript. This work was initiated when the first named author was visiting the Fields Institute in 2010. The support from the institute is gratefully acknowledged. ∗ Corresponding author Email addresses: [email protected] (Kyoung-Kuk Kim), [email protected] (Chi-Guhn Lee)

Preprint submitted to Energy Policy

May 16, 2012

In the fall of 2009, the Government of Ontario launched one of the most comprehensive FIT programs as part of the Green Energy and Green Economy Act. This paper was, in particular, motivated by the FIT program of Ontario, which have received positive responses from local residents and renewable energy developers (see Yatchew and Baziliauskas [22]). Despite the strong initial reaction from investors, its long-run performance and sustainability still remain to be seen. As such, the purpose of this paper is to evaluate the performance of a generic FIT program in the long-term. Specifically, we aim to investigate the interplay between a FIT policy structure and some possible constraints. As noted earlier, FIT programs could differ in their details across countries and local governments. Couture and Gagnon [6] discuss seven different payoff structures in FIT programs, dividing them into market-independent FIT policies and market-dependent FIT policies. Features that distinguish those payoff structures are, for instance, whether the payoff is fixed or related to the spot market electricity price, or whether it is inflation adjusted or not, etc. If it is accepted that the possibility of an individual or a developer to sign up for a FIT contract depends on its net present value (NPV), then different payoff structures would lead to policy dependent NPVs, resulting in different outcomes, in addition to the effects of regional inhomogeneity. Moreover, the determination of tariff levels should be taken into account because it is related to the total cost of a FIT program and this acts as an economic burden on ratepayers when the cost is distributed among them, thus, possibly leading to some political pressure on the government. Therefore, there are complex interactions between the expected performance of a FIT program and policy mechanisms. Hence, as a by-product, we examine how the optimal tariff levels and final payoffs to investors behave. For this, we work with a stylized model and payoff structures, and assume that program participants can freely transact in the deregulated spot electricity markets. Also, we focus on solar photovoltaic (PV) technology among several renewable energy technologies (RET). This paper is organized as follows. Section 2 reviews related works and Section 3 presents our model, the evaluation method, and constraints. In the following section, we report and discuss numerical results. Finally, Section 5 concludes the paper. 2. Backgrounds and Related Works Spreading the use of RET is a highly complex task that requires all of technology innovations, policy improvements, and market developments. Thus, policies for renewable energy sources (RES) have gone through many changes over time to reflect changes of relevant factors. The FIT program of Ontario, which is our motivating example, is also preceded by the Standard Offer Program (SOP) that started in 2006 by the Ontario Power Authority (OPA). The program was designed to encourage RET investments by paying premiums to small scale electricity generators, and it provided 0.42 CAD/kWh for solar energy and 0.11 CAD/kWh for biomass, hydro and wind, while the Ontario electricity consumers paid 0.09 CAD/kWh in 2007. However, the revenues in return for such investments were not enough, and failed to attract many small electricity producers. Considering that there were some collective movements among locals to install renewable energy production systems despite costs (RISE – Riverdale Initiative for Solar Energy 2006) and that there have been very strong reactions from people to the newly launched FIT programs, it is clear that the costs of RET systems have been the big hurdle for potential producers to take actions. More details are documented in Murray [16] and Yatchew and Baziliauskas [22]. The OPA [17, 18] documents small scale FIT program policies (less than 10 kW systems) for RET. It has different fixed price policies for different technologies, the solar PV being the highest. But, 20% of the fixed price for renewable energies except the solar PV escalates annually based on the Consumer Price Index. Under the Ontario FIT, all energy sources have 20 years contract term except waterpower which has 40 years. Each system can be directly connected to

2

the grid, between the load customer meter and the grid, or behind the load customer meter. This is a business contract between an investor and the OPA, i.e., the investor sells all produced electricity to the OPA at the fixed price. The FIT program is reviewed approximately every two years by the OPA. The Ontario FIT policy has the feature of fixed price guarantee combined with inflation adjustment, and the tariff level is adjustable at fixed time points. In the fast evolving electricity markets we see today, such a policy can provide stable conditions for investors. However, by fixing the price, it could also bring relative benefits or disadvantages to the administering entity and the usual ratepayers because other associated costs or economic conditions change, and such uncertainties could be quite a big burden. Taking into account regional specifics, there are several other policy options as analyzed in Couture and Gagnon [6]. To understand more fundamental differences, let us focus on four different FIT payment options. The first is fixed price tariff which is adopted by most countries, and it pays for the full amount produced by a system for a certain period specified in a contract (typically 15–20 years). The second FIT payment option is premium payment, meaning that an investor receives the total of the spot market electricity price plus a pre-specified premium. The third is hybrid payment in that the total payment is equal to the maximum of the spot price and a guaranteed price level set by the overseeing authority. The last is a variant of the second structure, in which the premium is a function of the spot price with a cap and a floor. The reason that we consider such variations is that a project’s NPV could be largely dependent on its payment structure, particularly in the current environment with changing spot electricity prices. Let us mention some existing works related to this paper. Klein et al. [11] analyze and discuss the FIT designs and practices employed by EU member countries. According to the authors, Germany and Spain have been very successful adopters of FIT policies, and FIT has been among the most frequently used instruments for the RES electricity generation support. However, detailed implementation differs across nations, and also the methods to determine the tariff levels are not same. For instance, Portugal determined the tariff level based on avoided costs, including climate environmental benefits and effects on the energy supply security, while most EU countries based the tariff levels on the generation costs such as initial investments, operating costs, inflation etc. Along this line of works, several papers address the issue of comparing different renewable energy promotion schemes. The earlier paper by the project Huber [10] documents and compares RES support schemes by EU countries. And Barbose et al. [1] present a comprehensive collection of information regarding solar PV programs in the US. The information includes systems, warranties, training programs, and incentive programs, among which warranties and incentives are related to our context. Comparison with schemes such as RPS, auction mechanism, TGC has been done in Butler and Neuhoff [3], Cory et al. [4], and Tam´as et al. [19]. More recently, Couture et al. [5] and Evans et al. [8] discuss FIT policy designs and more general market-based incentives, respectively. As explained above, our objective in this paper is to investigate the interactions between FIT policy structures, performance, and certain constraints in various scenarios. For this, we restrict our attention to one stylized model, four FIT payment options, and two plausible constraints on the government (or any entity that is responsible for the policy implementation). Our model largely draws upon the model presented in van Benthem et al. [20]. The authors analyze the California Solar Incentive program which consists of installation subsidy and tax credit. These incentives are designed to decrease annually and disappear in 2017. Although their work is not an analysis of a FIT policy, it provides important information regarding analysis methods, consumer choice modeling and solar PV cost reduction. We employ the first two of four main models from their work; consumer choice, cost reduction, environmental externalities, and economic efficiency as a policy goal. Firstly in the consumer choice modeling, the number of participants becomes a function of the NPV of a project as each potential FIT contract signer is motivated by the economic gain. In addition, the effect of social diffusion is considered as well in a specific format. Later, 3

this deterministic modeling is modified to reflect probabilistic aspects of consumer behavior. Secondly, note that FIT programs are designed to provide incentives to implement RET despite large initial investment costs. Therefore, the modeling of PV cost reduction becomes a crucial part of the analysis, and the learning-by-doing approach is adopted by van Benthem et al. [20]. In this approach, the total cost is determined by the local and the global number of total solar PV installations, with fixed learning rates. See Section 3 for modeling details. Before moving onto the next section, we mention three streams of research papers that are indirectly related. In electricity markets, electricity derivatives have been developed rapidly in recent years and valuation methods have also been an active area of academic research. One such paper by Deng and Oren [7] summarizes several important kinds of financial derivatives in the unregulated electricity market together with some distinctive features of the market different from the capital market. Due to optional features involved in the evaluation of FIT policies in our model, we apply the pricing theory of financial derivatives under certain assumptions that make the approach valid. For example, we assume that the risk neutral pricing mechanism is applicable and we compute the NPV after deriving risk neutral parameters in our model. We refer the reader to any existing asset pricing textbook for an introduction to asset pricing theory of financial products or electricity derivatives. On the other hand, there are many research works that deal with energy policies and projects from operations research perspectives. An interested reader can consult Mavrotas et al. [14] and Mu˜noz et al. [15], for instance. Lastly, recall several different FIT payoff structures introduced above. Those belong to the class of pricing policies that involve certain price guarantees, and such policies have also been the target of fruitful academic research. See Marcus and Anderson [13], Levin et al. [12], and references therein for more information. 3. The Model To investigate FIT programs in more details, we identify major ingredients of a typical program. As mentioned earlier, we focus on solar energy, in particular, electricity produced by solar PV systems. However, the underlying modeling approach in this paper should be extendable to other RET cases in a similar manner. The first determinant is, of course, the characteristics of solar PV systems such as energies produced, lifetime, maintenance, cost structure, etc. The second is the details of a FIT program, including program duration, contract period, and payoff structure. Those determine whether such a program is attractive economically to a potential participant. This part is also concerned with the calculation of the NPV of a contract. Then, the third component is a model for the consumer behavior, governing the law of how many will sign up based on the economic value of a contract. Because of a possible option feature of a FIT payoff structure and the stochastic nature of electricity price dynamics, we describe the price dynamics and an option valuation model as the fourth one. Lastly, we discuss plausible objectives and constraints from a policymaker’s point of view. 3.1. Solar PV This method of generating electricity uses solar cells based on the photovoltaic effect, converting sun light energy into electric currents. Even though solar cells are widely used already, e.g., portable calculators, our main focus is on household usage which accounts for a major part of the total electricity consumption. The capacity of a solar PV system is measured in watt peak (Wp), or DC watt output under the test condition of 1000W per square meter, 25◦ C, and air mass 1.51 . The energy produced by 1 kWp system is given in kWh per year, and it depends on the location 1 Air

mass is a measure of optical path length in terms of a ratio relative to the length at the zenith. Air mass 1.5 means solar zenith angle

48.19◦ s. (see http://rredc.nrel.gov/solar/spectra/am1.5)

4

average solar system size (x)

3.1 kWp 1314 (= 0.15 × 365 × 24)

location efficiency (η) PV life (L)

30 years

warranty life (W )

20 years

maintenance cost (m)

$240

Table 1: Solar PV characteristics

efficiency of the area where the system is installed. This is expressed as E (kWh/year) = x (kWp) ×η where η = (location efficiency) ×365 × 24. In our numerical study, we set this location efficiency equal to 0.15 which is close to the efficiency of Toronto, Canada (see Murray [16]). Here, x stands for the average solar PV system capacity. A solar PV system is assumed to have its lifetime equal to L, and this is often more than 30 years. Also, solar system providers might attach warranties to guarantee the system performance. Such a warranty applies to the whole system and its length is set equal to constant W in our model. In addition, in the period of W + 1 to L, the system incurs maintenance cost of m which grows annually at a risk free rate r. Table 1 records parameter values used in this paper. The investment in a solar PV system requires a significant capital for individual homeowners. However, the cost of installment is going to be reduced as new technologies are introduced and the market for solar cells matures. In detail, we split the cost into two parts, adopting the approach of van Benthem et al. [20]. The PV module cost takes up a half of the installation cost, which is rather the effect of global learning. The other part, so called the balance-of-system (BOS) cost, is due to regional specifics, hence affected by local learning process. This includes an inverter, PV array support structures, cabling, equipments, and installing. In van Benthem et al. [20], the total cost is given by −βBOS M ct = αM Q−β G,t−1 + αBOS Qt−1

where βM , βBOS are learning coefficients greater than zero and QG,t−1 , Qt−1 are cumulative installments (global and regional, respectively). Therefore, the cost is reduced further as installments or learning coefficients increase. Associated with learning coefficient βM , learning rate and progress ratio are defined by 1 − 2−βM and 2−βM , respectively. Note that progress ratio (and similarly for learning rate) represents how much the cost is reduced when the cumulative production is doubled. We use similar concepts for βBOS . Even though we treat local and global installments separately in our model, we note that this can be relaxed when the local market is large enough to have an impact on the global market. Since global learning happens worldwide, provincial FIT programs and the number of local installments do not ∑t−1 affect the first term of ct above. Rather, it is Qt−1 = Qinitial + i=0 Ni that is affected where Ni is the number of sign-ups in the period i and Qinitial is the number of installments prior to the start of a FIT program. Likewise, QG,initial stands for the initial size of global installments. The growth of the global installment size is given exogenously at the rate of gM . Thus, though period 0 to period t − 1, the size becomes QG,t−1 = QG,initial egM t . To determine appropriate parameter values, recall that global and local factors correspond to a half of the cost each. −βBOS M Hence, if c0 is the initial cost of the average solar PV system, then αM Q−β G,initial = c0 /2 and αBOS Qinitial = c0 /2.

These lead us to c0 c0 ct = exp (−βM gM t) + 2 2

( 1+

5

1

t−1 ∑

Qinitial

i=0

)−βBOS Ni

.

progress ratio for module (2−βM ) −βBOS

progress ratio for BOS (2

global solar growth rate (gM ) initial cost (c0 )

)

0.9 0.9 10% $35,000

Table 2: Parameter values for solar PV installation

Necessary parameter values are given in Table 2 except for Qinitial 2 . We determine this parameter by matching the growth rate of local installments to that of global installments. In particular, this is done in the case that the NPV of a FIT contract is zero, and thus it is not an economically attractive investment option for potential participants. This is further discussed below in Section 3.3. 3.2. Feed-in Tariff program Our model calculates on a yearly basis, meaning that one year is one period, and tariff levels and electricity prices are fixed at constant in each period. In period t, a certain number of new installments are made, following the consumer behavior model described below. These installments are done in period t and new FIT contracts become effective in period t + 1, ending in period t + T . Recall that a FIT contract is a business contract between an individual and the overseeing authority such that the former sells electricity generated by the installed solar PV system to the latter as specified in the contract during periods t + 1 to t + T . By allowing certain excess returns on such an investment, FIT programs aim to increase the size of solar PV installments and eventually the electricity generated from RET. Hence, FIT programs only last for a finite number of periods, say t = 0, . . . , M . We also note that it is possible that a FIT contract period is shorter than the lifetime of a solar PV system. In such a case, we assume that a participant sells the generated RES electricity to the spot electricity market at the prevailing price thanks to the provided connection to the grid. This clearly includes the case that she consumes the RES electricity for household use, reducing the amount of electricity that she purchases from the market at the spot price. The last item to be mentioned regarding a FIT contract is its payoff structure. Even though there could be many variants, we focus on four structures as they represent popular and major choices in implemented FIT programs. More details can be found, e.g., in Couture and Gagnon [6]. We use ft in period t to denote the decision variable of the administering entity, which affects the final FIT payoff. For an investor who signs up in period t, the future payoffs are denoted by Ht+1 , . . . , Ht+T and given as follows: for a unit quantity of energy produced (kWh) and for spot electricity price ps per kWh in period s ∈ {t + 1, . . . , t + T },   f,   t     max(ft , ps ), Hs =    ft + p(s ,   ))   max ps , ft + min (θ, ¯ max(ps , θ) . The first one is the fixed price model which is like a financial swap contract. The second is the same as the final payoff of the spot market gap model in which the FIT payment fills the gap when the spot price falls below the threshold specified in a contract. This can also be thought of as a minimum price guarantee. The third payment structure defines 2 The

initial cost is rather high compared to the current price due to the recent reduction of solar PV costs. However, the insights gained from the

model and the numerical tests should remain unchanged.

6

FIT program duration (M ) FIT contract length (T )

9, i.e., total 10 years 20 years

Table 3: Parameter values for a FIT contract

the payment as the sum of the market price and a fixed premium. Under this payment scheme, the profit to the investor can be a landslide when the market price of electricity increases. Therefore, the fourth payment structure has been suggested to reduce the premium as the electricity price goes above a pre-set level. This scheme is called a sliding premium. In many cases, there can be a cap and a floor in addition to the sliding premium. Other variants include variable premiums with fixed degradation rate, for example. The constants θ and θ¯ are included to represent such cap and floor features. If the price ps is lower than θ, then the payment is fixed at ft + θ. And if the price is moderately ¯ then the payment becomes ft + θ. ¯ Figure 3.1 shows the final payoffs Hs as a function of the spot price larger than θ, ps at time s. Some model specifics are given in Table 3. The important feature to insure a successful FIT program is the economic value of a contract, represented by its NPV. Due to the uncertainties inherent in future electricity prices, we take mathematical expectation of future cash flows adopting the financial option pricing theory. The expectation is calculated under a risk neutral (rather than physical or real world) probability measure P of which existence is guaranteed if the electricity market is efficient and so no arbitrage condition holds. Therefore, the NPV of a FIT contract in period t can be calculated as ψt = xη

t+T ∑

L L [ ] [ ] ∑ ∑ E e−r(s−t) Hs |Ft + xη E e−r(s−t) ps |Ft − e−r(s−t) ms − ct

s=t+1

= xη

t+T ∑

s=T +1

s=W +1

[ ] E e−r(s−t) Hs |Ft + xη(L − T )pt − (L − W )m − ct .

s=t+1

The conditional expectation E is taken under P where Ft is the set of all available information at time t. In the risk neutral probability measure P, asset price processes with a risk free discount rate r are martingales. Therefore, the expectation of the second term, which represents the time t value of the RES electricity after the FIT contract ends, is equal to the time t price of electricity in the spot market. In the third term, ms represents the maintenance cost in period s, which is assumed to be m · er(s−t) for simplicity. As for the first term, we need an option valuation model and it is the topic of a later subsection. Remark In practice, the minimum price guarantee type of payoff structure can be implemented in different ways. In case the developer is responsible for selling the electricity at the spot price higher than ft , the developer will be exposed to risks such as the risk of not being able to sell in some periods. In such cases, she may want to have a long-term contract with a utility company to receive payments possibly less than the spot price but still higher than ft . The proposed model in this paper can still be used with the spot price replaced by the spot price minus a fee paid to the utility company. Alternatively, the overseeing authority might guarantee the payment to be the larger of the spot price ps and the guaranteed price ft . In either scenario, the NPV can be computed in the exactly same way and hence our analysis applies with additional fees if any. Remark Although our model is yearly based, the analysis can be done at any time scale, in which the features like monthly electricity price dynamics or correlation between the spot price and the stochastic generation pattern of the solar PV can be incorporated as well. This can be done by adding more nodes to the lattice constructed below, or by modeling the generation pattern as a second stochastic factor. 7

FIT payout = price guarantee

FIT payout min price guarantee

f

t

ft

ps (price)

ft

ft+θ

ps (price)

FIT payout

FIT payout f +θ t

premium

premium ft

ft

θ

ps (price)

θ

ps (price)

Figure 3.1: Four payment structures: (upper left) fixed tariff (upper right) minimum price guarantee (lower left) fixed premium (lower right) sliding premium with cap and floor.

8

initial demand parameter (a0 )

1000

sensitivity parameter (b)

0.0006

maximum annual market size (Nmax )

200,000

diffusion parameter (γ)

0.15

Table 4: Parameter values of a choice model

3.3. Consumer choice The number of new sign-ups in period t, denoted by Nt , is governed by several factors. Even though economic motivation could be the most important driver in the consumer decision making process, it is not the sole reason as an individual signs the contract out of environmental awareness or the popularity of the program if it successfully spreads through local communities. Nor does everyone participate even if the NPV of a contract is positive. All of these are described in the following model, which is essentially a stochastic version of the consumer choice model in van Benthem et al. [20]. The probability that a person signs a new contract in period t = 0, . . . , M is given by ζt =

at , at + (Nmax − at ) exp(−bψt )

t≥0

where Nmax is the maximum possible annual sign-ups and at ≤ Nmax . Parameter b measures how sensitive a consumer is to the NPV of a FIT contract. Even if the FIT contract does not provide any expected profit (ψt = 0), the probability can still be positive with ζt = at /Nmax . Thus, at can be thought of as the ratio of people who are willing to sign up for reasons other than economic benefits. But, of course, limψt ↓−∞ ζt = 0 and limψt ↑∞ ζt = 1. In addition to new sign-ups that come from a pool of size Nmax , a number of signers come outside the pool. For example, a person who did not sign up last year might reconsider it this year based on the experience of neighbors. This kind of effects is captured by a technology diffusion term dt and the evolution of diffusion is basically a quadratic function ranging between 0 and Nmax : for t ≥ 1, dt = γNt−1 (1 − Nt−1 /Nmax ) . On the other hand, the demand parameter at itself is dynamic and dependent on the diffusion of solar PV to incorporate the knowledge effects about the financial and non-financial benefits of renewable energies. For this, it is set as at = at−1 (1 + dt−1 /Nt−1 ) for t ≥ 1. Finally, the number of new sign-ups in period t is the result of individuals’ decision making modeled by i.i.d. Bernoulli random variables and the technology diffusion: for t ≥ 0, Nt =

N∑ max

Xi + d t ,

P(Xi = 1) = ζt .

i=1

In period 0, diffusion term d0 is set equal to zero. We give parameter values in Table 4. Remark With initial local installments Qinitial , the expected number of total installments in period 0 becomes E[Q0 ] = Qinitial + E[N0 ] = Qinitial + a0 if ψ0 = 0, where a0 is the number of new installments in period 0. We assume that this number matches Qinitial egM where the growth rate of total installments is equal to the global growth rate. Thus, Qinitial = a0 /(egM − 1). 3.4. Option valuation

[ ] In this subsection, we describe the option valuation model for the computation of E e−r(s−t) Hs |Ft . Our model is

a regime switching model with three regimes. A regime switching model is widely used in pricing of financial options because it provides a sound economic interpretation by modeling ups and downs of economies and also because it 9

accounts for time-varying volatilities that the Black-Scholes option pricing model lacks. A lattice version of a regime switching model was studied by Bollen [2] and its multi-regime version by Wahab and Lee [21]. Let us briefly explain the model and the option valuation procedure. Suppose that there are n regimes, in each of which the log-return of electricity price pt follows a normal distribution with mean µ ˆi and standard deviation σ ˆi . The state of regime in period t is denoted by st and regime switching from regime i to regime j occurs with transition probability ξij = P(st = j|st = i) for i, j = 1, . . . , n, independently of ∑n the return distribution of electricity price. Of course, j=1 ξij = 1 for each i. At each node of a lattice, there are 2n + 1 branches attached, two for up and down case for each regime and one middle branch shared by all regimes. Thus, in ( each ) regime, there are three possibilities(of price ) movements (from price pt in period t), namely higher price pt exp ϕˆi , same price pt , or lower price pt exp −ϕˆi for some positive rate of return ϕˆi , simplifying the assumption √ ˆi2 h + µ ˆ2i h2 where h is the time step size of a lattice. that the log-return is normally distributed. Each ϕˆi is given by σ However, price moves constructed as such make the lattice non-recombining and this results in the exponentially increasing number of branches, thereby making an implementation impractical. To avoid this problem, we need to go through the following adjustment. First, we sort the ϕˆi in an increasing order, using the same symbols, i.e., ϕˆ1 < · · · < ( ) ϕˆn . We re-arrange the µ ˆi and the σ ˆi accordingly, taking the hats off. Then, we set ϕ = max ϕˆ1 , ϕˆ2 /2, . . . , ϕˆn /n and define ϕi = iϕ for each i. As for the probabilities associated with price movements, they are given to match µi and σi2 by looking at first two moments of log-returns. If we denote up, middle and down probabilities in regime i by πi,u , πi,m , and πi,d , respectively, then we have πi,u ϕi + πi,m × 0 + πi,d (−ϕi ) = µi h, 2

πi,u ϕ2i + πi,m × 02 + πi,d (−ϕi )2 − (µi h) = σi2 h, πi,u + πi,m + πi,d = 1. The solution to this system of equations is ( ) µi h 1 ϕˆ2i + πi,u = , 2 ϕ2i ϕi

πi,d

1 = 2

(

µi h ϕˆ2i − ϕ2i ϕi

) ,

πi,m = 1 − πi,u − πi,d .

Suitable constraints on parameters ensure that πi,u and πi,d are indeed probabilities with values in the interval [0, 1]. Also, πi,u + πi,d = ϕˆ2 /ϕ2 is in [0, 1] as well due to the definition of ϕi . Hence, πi,m becomes a legitimate probability. i

i

Now, given such a lattice, let us describe the option valuation procedure. With a payoff Hs in period s, we compute each realized payoff, say C(s, i, j) at each of 2ns + 1 nodes in period s. Here, i denotes the corresponding regime and j the index of the node, ranging from 2ns to −2ns. Given C(t + 1, i, j) values for all i, j, the backward induction step for the computations of the prices in period t is −rh

C(t, i, j) = e

n ∑

( ) ξik πi,u C(t + 1, k, j + i) + πi,m C(t + 1, k, j) + πi,d C(t + 1, k, j − i)

k=1

for i = 1, . . . , n and j = −2nt, . . . , 2nt. The induction ends when we reach time 0, the valuation point. Note that h is 1 in our model. This valuation approach requires the knowledge of the current regime which we assume is feasible from available information in the markets such as financial indices, interest rate policies of a government, exchange rates, and so on. The consumer choice model described in Section 3.3 depends on the spot electricity price. (The level of dependency is governed by the parameter b.) And this electricity price changes stochastically over time, the feature that we 10

incorporate by a regime switching model. On the other hand, the option valuation is done under a risk neutral probability P. In this probability measure, any financial contract has a mean return equal to a risk free rate r. Therefore, in our numerical implementation, we firstly generate time varying prices under the physical measure, and secondly we compute the conditional expectations under a risk neutral measure. The former is done under a lattice with some mean and volatility parameters, and the latter is done under a lattice with mean equal to r and the same volatility parameters. 3.5. Objectives and constraints There are many reasons why jurisdictions implement FIT programs, ranging from energy security, concerns about climate change to job creation, as well as peak shaving. The success of such implementations is determined by the number of sign-ups, which is proportional to the probability of sign-up ζt in each period. Consequently, we aim to configure program details to maximize ζt . And, in the definition of ζt , we note that this probability increases as the NPV ψt increases, which in turn depends on the decision variable ft . Therefore, we should choose the maximum possible ft that utilizes available financial resources for a program. Recall that FIT programs induce the participation of residential households or developers in renewable energy markets, by providing some reasonable returns on investments. In the case of Ontario, the tariff levels for RET are multiples of the current electricity price, that of solar being the highest. Certainly, such additional financial rewards do not come free, and often the responsible authorities charge extra fees on the usual ratepayers, say by Bt in each period. However, such an increase of electricity bills might result in the high risk of inflation as well as conflicts of interest. Moreover, it is entirely possible that there might have been stronger responses than expected in previous periods, say 1, . . . , t − 1. If such additional expenses are to be paid by ratepayers in period t and beyond, then the resulting electricity bill increases could be unacceptable. Since the interaction of policy structures and economic constraints is the main target of this paper, we estimate the household burden (possibly very large) and hope to gain deeper understanding of the problem, rather than abiding by the constraints below strictly. There are many different ways of setting such constraints. Among many plausible scenarios, we choose two scenarios that are described shortly. Both are somewhat between myopic and far-sighted. Suppose that we are in period t, and there were Ni number of participants in period i for i = 1, . . . , t − 1. Let us denote the payoff to a participant who signed up in period i by Hi,s for period s = i + 1, . . . , i + T . The first one is concerned with the budget balance between the remunerations to participants N1 , . . . , Nt in the current period t and the next period t + 1, and the revenues from the electricity generated by (to-be-)installed systems as well as the additional electricity bill payments by ratepayers in periods t and t + 1. Then, the constraint can be expressed as follows: t−1 ∑

[ ] [ ] xηNi E Hi,t + e−r Hi,t+1 |Ft + xηE e−r Nt Ht,t+1 |Ft

i=1

≤2

t−1 ∑ i=1

xηNi pt + xηE[Nt |Ft ]pt +

t−1 ∑

(constraint I) (Bi + e−r Bi )K + e−r Bt K.

i=1

{ } On the right side of the inequality, the modeling assumption that e−r(s−t) ps s≥t is a martingale under P is used

to get xηNi E[pt + e−r pt+1 ] = 2xηNi pt . Lastly, we assumed that the bill increase Bt for period t is received in s = t + 1, . . . , t + T and that the total number of ratepayers is K. We note that the only decision variable in the equation is ft which affects Ht,s and Nt . Other numbers Ni and Hi,s for i < t are already simulated or determined. In the second scenario, we consider the remunerations to participants N1 , . . . , Nt−1 only in period t (without period t + 1). Also, this year’s revenues from RES electricity and Bi ’s are included. However, regarding the (random)

11

Nt number of participants, the total expected remunerations and revenues enter the constraint: ] [ t+T t−1 ∑ ∑ xηNi Hi,t + xηE e−r(s−t) Nt Ht,s Ft s=t+1 i=1 (constraint II) t−1 t+T ∑ ∑ (xηNi pt + Bi K) + T xηE[Nt |Ft ]pt + e−r(s−t) Bt K. ≤ s=t+1

i=1

Note that we used the martingale property of discounted electricity prices in the second term of the right hand side of the inequality. As argued in the next section, the effects of constraints are somewhat mixed, and it is potentially quite interesting to see what kind of policy constraints would be desirable. However, this is beyond the scope of this paper and thus we leave it as a future research topic. 4. Numerical Studies We run several numerical experiments to study the effects of different payoff structures, constraints, and economic outlook on the performance of a FIT program. In addition, we conduct some sensitivity analysis. A FIT policy depends on many parameters. It is not our objective to go through comprehensive sensitivity analysis with respect to every single parameter. Instead, we focus on price transition matrix Γ, the NPV sensitivity parameter b, target household burden B, and the diffusion parameter γ. The reason for this choice is that they represent some important characteristics of target communities. However, in our framework, it should be quite clear how to incorporate additional sensitivity analysis for other parameters such as volatility of electricity prices or interest rate changes. The basic tool of our analysis is simulation. We randomly generate 1,000 scenarios of future electricity prices, following the regime switching model described above. Then, for each scenario, we maximize our objective function, i.e., the expected number of sign-ups over the duration of a FIT program under a chosen constraint. Our decision variable is the schedule of {f0 , . . . , fM }, which affects the NPV of the FIT program in each year and, in turn, the sign-up probability. Lastly, we compute the arithmetic averages of relevant measures in the 1,000 scenarios. Throughout the numerical studies, we set the total number of households K to be 1,000,000 and use the parameters as presented in Tables 2, 3 and 4. For each price path, we numerically search an interval ($0.001, $2) for an optimal level ft in year t so as to maximize the objective function under a given constraint. Due to the nature of uncertain electricity prices, the search may not be successful, in which case we set ft as the best possible value which yields the estimate of household burden close to but less than the target household burden under the given policy. In the worst case, there can be no ft that satisfies both the given constraint (type I or II) and the household burden being less than the promised cap. Then, we set ft to a value in the search range by letting the burden exceed the target but yet enforcing the given constraint to be honored. After finding ft , the number of new sign-ups in period t is simulated. Thus, each price path leads to an optimal ft schedule, estimates of household burden, and the numbers of new sign-ups during the FIT program period. Below we give parameter values that have not been specified so far. Three regimes in the regime switching model are assumed to have the parameter values in Table 5 for return distributions. The higher µi is, the average electricity price tends to move higher in the future. On the other hand, when σi is higher, the electricity price fluctuates more, yielding higher uncertainties in the future prices. For option valuation, we need to replace the rate of return in the regime switching model with a risk free rate r, which is assumed to be 3% in this paper. Since we are working on a long-term contract expanding more than 20 years, time-varying interest rates are more plausible. However, we do not consider such an extended version of the model because it is a difficult task to predict long term future interest rate moves (which requires separate substantial studies) 12

µ1

µ2

µ3

σ1

σ2

σ3

case 1

1%

3%

5%

20%

30%

40%

case 2

1%

3%

5%

40%

50%

60%

case 3

1%

3%

5%

60%

70%

80%

Table 5: Different parameter sets for future price scenarios

and also because our focus is on policy comparisons due to different option features. Policy interactions with interest rate changes remain to be studied in the future. In the sensitivity analysis below, we vary parameters b, B, and γ in addition to Γ = (γij ). More specifically, we take three values of b = 2 × 10−5 , b = 4 × 10−5 , and b = 6 × 10−5 , then we compare the FIT performances. From our consumer choice model, note that the sign-up probability increases as this parameter b increases. As for B and γ, we also consider three cases, (20, 40, 60) and (0.05, 0.15, 0.25), respectively. The final parameter of interest is the transition matrix which governs the regime dynamics. The following three different cases are considered in our experiments: 

0.8 0.15 0.05

 ΓL =  0.3

0.6

0.3

0.4



 0.1  , 0.3



0.6

 ΓM =  0.1 0.1

0.3 0.1



 0.8 0.1  , 0.3 0.6



0.3

 ΓH =  0.1 0.05

0.4

0.3



 0.3  . 0.15 0.8 0.6

These different transition matrices can be understood as representing different outlooks for the electricity markets in the future. For example, ΓL is the case that electricity prices tend to stabilize and ΓH the case that there is a higher probability of electricity prices moving to the high volatility regime. 4.1. Effects of payoff structure, constraint, and economic outlook In this subsection, we set the NPV sensitivity parameter b = 6 × 10−5 , target annual household burden $40, and the case 1 where three regimes of the electricity price have volatilities specified as 20%, 30%, and 40%. We, then, look at how the payoff structure affects the performance of a FIT program under each of three regime transition matrices. Recall that we have four payoff structures; fixed price, minimum price guarantee, fixed premium, and sliding premium. We denote them by ps1, ps2, ps3, and ps4 in this section, respectively. The two constraints introduced in the previous section are written as const1 and const2. Figure 4.1 shows the expected number of sign-ups for the duration of the FIT program under const1. By comparing three figures, we also see how differently each FIT policy performs according to regime transition matrix. From the results, we observe that ps2 outperforms all others, and ps3, ps4 are more or less similar. The worst performer is the fixed price policy ps1 and the performance gap between ps1 and others increases as we move from ΓL to ΓH . This can be understood as the result of the optional feature in ps2, ps3, and ps4. As the electricity price becomes more volatile, the real option value in the FIT policy increases, and this leads to more positive responses from potential developers. However, the best performance of ps2 is not without penalty. As shown in Figure 4.2, ps2 has the largest estimates of household burden. This could be a serious problem to an energy authority who oversees the FIT policy implementation. Figure 4.3 reports the total remuneration per kWh paid to the developers who sign up in each period (final income to a developer from tariff and/or electricity price) under four policy structures. The level of remuneration decreases over time, and the chosen regime transition matrix affects the sizes of the remunerations. One notable result is that the

13

ΓL

4

6

x 10

ps1 ps2 ps3 ps4

4

3

2

1

0

x 10

ps1 ps2 ps3 ps4

5

number of sign−ups

number of sign−ups

5

ΓM

4

6

4

3

2

1

1

2

3

4

5

6

7

8

9

0

10

1

2

3

4

year

5

6

7

8

9

10

year ΓH

4

6

x 10

ps1 ps2 ps3 ps4

number of sign−ups

5

4

3

2

1

0

1

2

3

4

5

6

7

8

9

10

year

Figure 4.1: Number of sign-ups with b = 6 × 10−5 , B = 40, case 1, constraint I.

remuneration level of ps4 gets smaller than that of ps3. This coincides with the typical observation that the sliding premium gains more acceptance among public by preventing over-support for developers. In our experiments, we set the cap for premium at 0.6 and the floor at 0.24. Although not reported here, it turns out that the performance of ps4 is sensitive with respect to the cap and the floor levels. By choosing a wrong set of cap/floor levels, we lose this nice feature of sliding premium. We can also conduct the same set of tests under const2. One case of performance comparison and estimates of household burden is presented in Figure 4.4. However, note that all policy structures except the fixed price ps1 behave quite similarly. This result can be seen as the result of some intrinsic property of the financial constraint and we lose the way of exploiting different policy mechanisms if we choose our decision variable ft under such a constraint. The last observation here is that estimated household burdens are kept at the target level B = 40 only for ps3 + const1 and ps1 + const2. Other cases re-direct part of the cost of a program to ratepayers, increasing the household burden over the target level B. There is yet another measure to be considered when we compare policy structures. We can calculate the total capacity induced by a FIT program after the numbers of sign-ups are simulated under each scenario (we have 1,000 price scenarios). On the other hand, we can consider the total cost that was necessary to support this FIT program. The computation of the total spending consists of two parts. The first is the remunerations from the FIT program to the program participants and the second is the electricity amount generated by installed systems (multiplied by spot market price), which is gain rather than spending. Then, by dividing the total capacity by the total spending, we can calculate the induced capacity per dollar under different policy structures. The results are reported in Table 6. The ratios for combinations of payoff mechanisms, constraint types, and transition matrices are ranging between 3.00 to 14

ΓL

ΓM

120

120 ps1 ps2 ps3 ps4

110 100

100 household burden

90

80 70 60

80 70 60

50

50

40

40

30

30 1

2

3

4

5

6

7

8

9

20

10

1

2

3

4

5

year

6

7

8

year ΓH 120 ps1 ps2 ps3 ps4

110 100 90 household burden

household burden

90

20

ps1 ps2 ps3 ps4

110

80 70 60 50 40 30 20

1

2

3

4

5

6

7

8

9

10

year

Figure 4.2: Estimated household burden ($) with b = 6 × 10−5 , B = 40, case 1, constraint I.

ΓL

ΓM

ΓH

Mean

ps1

const1

3.682

4.031

4.480

4.064

ps2

const1

3.397

3.363

2.967

3.242

ps3

const1

3.021

3.002

2.758

2.927

ps4

const1

3.291

3.287

2.954

3.177

ps1

const2

3.154

3.294

3.515

3.321

ps2

const2

3.143

3.198

2.995

3.112

ps3

const2

3.120

3.170

2.968

3.086

ps4

const2

3.133

3.183

2.975

3.097

Table 6: Ratio of induced capacity versus total FIT remunerations with b = 6 × 10−5 , B = 40, and case 1

15

9

10

ΓL

ΓM ps1 ps2 ps3 ps4

1.1

1 FIT remuneration

FIT remuneration

1 0.9 0.8

0.9 0.8

0.7

0.7

0.6

0.6

0.5

ps1 ps2 ps3 ps4

1.1

1

2

3

4

5

6

7

8

9

0.5

10

1

2

3

4

5

year

6

7

8

9

10

8

9

10

year ΓH ps1 ps2 ps3 ps4

1.1

FIT remuneration

1 0.9 0.8 0.7 0.6 0.5

1

2

3

4

5

6

7

8

9

10

year

Figure 4.3: Total FIT remuneration ($/kWh) with b = 6 × 10−5 , B = 40, case 1, constraint I.

ΓM

4

4

x 10

ps1 ps2 ps3 ps4

3.5 3

ps1 ps2 ps3 ps4

110 100 90 household burden

number of sign−ups

ΓM 120

2.5 2 1.5

80 70 60 50

1 40 0.5 0

30 1

2

3

4

5

6

7

8

9

20

10

year

1

2

3

4

5

6

7

year

Figure 4.4: Number of sign-ups and estimated household burden ($) with b = 6 × 10−5 , B = 40, case 1, constraint II.

16

ΓL

ΓM

ΓH

Mean

ps1

const1

3.091

3.279

3.460

3.276

ps2

const1

3.498

3.998

4.872

4.123

ps3

const1

3.968

4.874

6.920

5.254

ps4

const1

3.630

4.344

5.643

4.539

ps1

const2

3.270

3.560

4.215

3.682

ps2

const2

3.189

3.104

2.474

2.922

ps3

const2

3.233

3.136

2.488

2.953

ps4

const2

3.207

3.113

2.474

2.931

Table 7: Ratio of induced capacity versus ratepayers’ spending with b = 6 × 10−5 , B = 40, and case 1

4.48. Overall, ps1 induces bigger capacities than other payoff structures. It is also observed that ps2–ps4 produce less and less capacities under unstable market conditions while ps1 produces more, which can be understood as the effects of optional features. Now from the viewpoint of ratepayers, one can calculate the total spending as the total additional payments made by ratepayers to support the FIT program. The results are reported in Table 7. Then, one can see the numbers are quite different from Table 6. In this analysis with cosnt1, non-fixed-price payoffs lead to bigger capacities and do more so under unstable market conditions. But, under const2, we do not observe such behaviors and the performances of ps2–ps4 are quite similar. From these observations, the performance of a FIT program is seen to depend on payoff structure, constraint considered, future economic outlook, or policymaker’s standpoint. 4.2. Sensitivity analysis In this subsection, we perform some numerical tests about how sensitive the outcomes of a FIT program are to some parameters. We restrict our attention to the following three parameters: target household burden B, NPV sensitivity parameter b, and diffusion parameter γ. In particular, the last two parameters represent consumer characteristics such as risk-averseness and acceptance of a new technology within communities. For the first one, the results are depicted in Figure 4.5 where we use b = 6 × 10−5 , case I, constraint I, and B = 40 as the base case. The upper row shows how many new sign-ups are observed when B = 20 compared to the case of B = 40 under two different transition matrices, ΓL and ΓH . We see that decreased household burden induces worse performance in the beginning, but the effect diminishes as time passes by. The lower row is when we compare B = 60 vs. B = 40. Obviously, there are now positive effects when we have a larger B in early periods, but the same pattern of diminishing effect appears. Comparing ΓL and ΓH , we observe that FIT performances are less fluctuating when the electricity prices tend to stabilize. The case of sensitivity parameter b is different from B and the behavior is more complex. Using b = 4 × 10−5 as the base case (together with other parameters as done in the previous paragraph), the upper row of Figure 4.6 shows the ratio of numbers of sign-ups when b = 2 × 10−5 and b = 4 × 10−5 . The case of b = 6 × 10−5 vs. b = 4 × 10−5 is shown in the lower row. Quite intuitively, when a customer is more sensitive (i.e., higher b) to the NPV of a FIT policy, it induces higher probability of sign-up when there is a reasonable return on investments. What is interesting is that this feature is persistent throughout the duration of the FIT program. Actually, it induces higher sign-ups toward the end of the horizon under ΓL . In the case of ΓH , the consequence is more dramatic, with big moves in the middle of the whole duration. We see a similar behavior for the diffusion parameter γ. As shown in Figure 4.7, when we have an increased

17

ΓL

ΓH ps1 ps2 ps3 ps4

0.9

ps1 ps2 ps3 ps4

0.95 ratio of number of sign−ups (20 vs 40)

ratio of number of sign−ups (20 vs 40)

0.95

0.85 0.8 0.75 0.7 0.65

0.9 0.85 0.8 0.75 0.7 0.65

0.6

0.6

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

year ΓL

8

9

10

1.5 ps1 ps2 ps3 ps4

1.4

ratio of number of sign−ups (60 vs 40)

ratio of number of sign−ups (60 vs 40)

7

ΓH

1.5

1.3

1.2

1.1

1

0.9

6 year

1

2

3

4

5

6

7

8

9

1.3

1.2

1.1

1

0.9

10

ps1 ps2 ps3 ps4

1.4

1

2

3

4

year

5

6

7

8

9

10

year

Figure 4.5: Ratio of number of sign-ups with b = 6 × 10−5 , case 1, constraint I: (upper) B = 20 vs. B = 40 (lower) B = 60 vs. B = 40.

diffusion parameter, the effects are, first, the number of sign-ups increases compared to the based case γ = 0.05, and second, the ratio of numbers of sign-ups when γ = 0.15 and γ = 0.05 increases over time. Therefore, we can conclude that a higher diffusion parameter more encourages new sign-ups and it is even more so toward the end of the FIT period. However, the magnitude of such increase is smaller than the case of sensitivity parameter b. Although we do not report in this section, we ran additional tests to see the effects of the parameters µ and σ, i.e., case 1 – case 3. Obviously, it is expected that the number of sign-ups would increase as the mean and the volatility levels increase, and this is what happens. Such behavior is more noticeable when there is an optional feature. Indeed, in case 3, the parameter values are so large that the probability of sign-up is almost 1, leading to the number of sign-ups hitting the maximum level each year. However, such a case is clearly non-implementable because the financial burdens on ratepayers and the overseeing authority would then be prohibitive. 5. Concluding Remarks In this paper, we presented a stochastic model for the evaluation and optimization of FIT policies. We particularly focused on four different payoff structures; first, an investor receives fixed price per electricity produced by RET, second, the maximum of a guaranteed price and the spot electricity price, third, a certain premium on the top of the spot price, and last, the spot price plus a sliding premium with a cap and a floor. Even though actual policy implementation depends on regional particulars, those four simplified options represent the main characteristics of FIT policies, i.e., whether a policy is market dependent, independent, or hybrid. Due to the optional features of FIT payment types, the NPV of a FIT contract is option-like; the volatility of the 18

ΓL

ΓH

0.8

0.8 ps1 ps2 ps3 ps4

0.7 0.65 0.6 0.55 0.5 0.45 0.4

ps1 ps2 ps3 ps4

0.75 ratio of number of sign−ups (2 vs 4)

ratio of number of sign−ups (2 vs 4)

0.75

0.7 0.65 0.6 0.55 0.5 0.45

1

2

3

4

5

6

7

8

9

0.4

10

1

2

3

4

5

year ΓL

7

8

9

10

ΓH ps1 ps2 ps3 ps4

1.65 1.6

ps1 ps2 ps3 ps4

1.7 ratio of number of sign−ups (6 vs 4)

1.7 ratio of number of sign−ups (6 vs 4)

6 year

1.55 1.5 1.45 1.4 1.35 1.3 1.25 1.2

1.65 1.6 1.55 1.5 1.45 1.4 1.35 1.3 1.25 1.2

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

year

6

7

8

9

10

year

Figure 4.6: Ratio of number of sign-ups with B = 40, case 1, constraint I: (upper) b = 2 × 10−5 vs. b = 4 × 10−5 (lower) b = 6 × 10−5 vs. b = 4 × 10−5 .

ΓL

ΓH 1.3

ps1 ps2 ps3 ps4

1.25 1.2

ratio of number of sign−ups (0.15 vs 0.05)

ratio of number of sign−ups (0.15 vs 0.05)

1.3

1.15 1.1 1.05 1 0.95 0.9

1

2

3

4

5

6

7

8

9

1.2 1.15 1.1 1.05 1 0.95 0.9

10

year

ps1 ps2 ps3 ps4

1.25

1

2

3

4

5

6

7

8

9

10

year

Figure 4.7: Ratio of number of sign-ups with b = 6 × 10−5 , B = 40, case 1, constraint I: diffusion parameter 0.05 vs. 0.15.

19

spot electricity price affects the NPV, and thus the consumer response to the program. The valuation method uses the well established theory of asset pricing in financial economics. Based on a regime-switching model with three regimes, we generate future scenarios of electricity prices and, at each time grid point, we calculate the NPV of a FIT contract under a risk neutral measure. From the viewpoint of a policy maker, the expected total cost of FIT programs matters as well. Since often such costs are distributed among the usual ratepayers, we considered two constraints which weigh the expected costs against the revenues from the ratepayers, differing on whether such a consideration is somewhat short-term or long-term. From numerical experiments, we noticed that there exists no single policy dominant structure in the sense of the number of sign-ups, the burden on ratepayers, and total capacity installed. Clearly, there is a general tendency that the policy has a higher number of sign-ups under a more volatile market, but this is at the cost of higher burden on ratepayers, which sometimes becomes excessive. When an economic constraint is considered as well, it adds an additional complexity to the whole picture. The optimal policy, therefore, depends on the policy objective, the method of burden sharing, and the forecasts of the future spot electricity prices. In this sense, our model could be a potentially useful tool although it is only among the first of such stochastic models to analyze energy policies which include option-like features. Certainly, more empirical testing, including empirical verification of model ingredients such as regime switching and cost reduction by learning-by-doing, remains to be done in the future. References [1] Barbose, G., R. Wiser, and M. Bolinger (2008) Designing PV incentive programs to promote performance: A review of current practice in the US, Renewable & Sustainable Energy Reviews, 12, 960–998. [2] Bollen, N. (1998) Valuing options in regime-switching models, Journal of Derivatives, 6, 38–49. [3] Buler, L. and K. Neuhoff (2008) Comparison of feed-in tariff, quota and auction mechanisms to support wind power development, Renewable Energy, 33, 1854–1867. [4] Cory, K., T. Couture, and C. Kreycik (2009) Feed-in tariff policy: design, implementation, and RPS policy interactions, Technical Report, National Renewable Energy Laboratory, Colorado, USA, available at http: //www.nrel.gov/docs/fy09osti/45549.pdf [5] Cory, K., T. Couture, C. Kreycik, E. Williams (2010) A policymaker’s guide to feed-in tariff policy design, Technical Report, National Renewable Energy Laboratory, Colorado, USA, available at http://www.nrel. gov/docs/fy10osti/44849.pdf [6] Coutuer, T. and Y. Gagnon (2010) An analysis of feed-in tariff remuneration models: Implicatoins for renewable energy investment, Energy Policy, 38, 955-965. [7] Deng, S. J. and S. S. Oren (2006) Electricity derivatives and risk management, Energy, 31, 940–953. [8] Evans, A., M. Gamez, M. Croucher, T. James (2011) Market-based incentives, Technical Report, Arizona State University, available at http://azsmart.org/papers/ [9] Gallego, G. and G. van Ryzin (1994) Optimal dynamic pricing of inventories with stochastic demand over finite horizon, Management Science, 40, 999–1020.

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