Faculty of Economics & Business (FEB) MSc. Economics Track: Markets & Regulation Martijn Heijnis 10410821 Wednesday, July 11, 2018 MSc. Thesis
Modelling prosumer investment behavior: an analysis of the proposed
regulatory reform in the Dutch retail electricity market
Supervisor: Dr. A.P. Kiss Abstract The Dutch government has announced the replacement, in 2020, of the net-metering incentive scheme for electricity prosumers. It is to be replaced by a feed-in subsidy for every kWh fed into the grid. The proposed subsidy will be calculated using a simple payback-period analysis. I develop a model to study PV investment behavior based on the assumption that population characteristics of attitude towards PV investment are log-normally distributed. This model predicts that setting the feed-in subsidy using a simple payback period method will lead to an overstimulated PV market in 2020, and a decline afterwards. I propose a more complete and refined method of calculating the optimal feed-in subsidy. I find that for any desired level of PV capacity in 2030, there exists a subsidy path that will ensure a smooth transition against limited costs for the regulator. Simulations of different scenarios are performed to demonstrate the predicted effects.Statement of Originality This document is written by Student Martijn Heijnis who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Table of Contents 1 Introduction ... 4 2 Dutch retail electricity market ... 7 3 Literature review ... 11 4 The Model ... 15 4.1 Modelling market quantities ... 15 4.2 Baseline theoretical model ... 18 5 Data and model inputs ... 22 5.1 Usage and generation profiles ... 22 5.2 Cost of generation: levelized cost of electricity (LCOE) ... 28 5.3 Market prices ... 35 5.4 Capacity limitations and current level of PV-adoption ... 38 6 Application ... 42 6.1 Simple cost analysis ... 43 6.2 Model calibration ... 51 6.3 Effect of the proposed subsidy path ... 58 6.4 Maintaining net-metering ... 62 6.5 Alternative subsidy paths ... 63 7 Results, limitations and extensions ... 70 7.1 Discussion of results ... 70 7.2 Limitations to model and its implications ... 74 7.3 Possibilities for extension ... 76 8 Conclusion ... 80 9 References ... 83 10 Appendices ... 88
1 Introduction The need for revision of the Dutch renewable energy incentive scheme for households has been the issue of debate. The Dutch government has committed itself to decrease CO2 emissions with 49% by 2030 (Wiebes, 2018): An ambitious goal that requires rethinking many old certainties and customs. However, the subject has not been approached from an economic-modelling perspective. The current tendency is to approach the PV investment decision behavior in terms of a simple payback analysis. I show that, although the per-kWh subsidy will be lower under the new subsidy scheme than under the current incentive scheme, this proposed subsidy has a highly distortionary effect. It causes extreme overstimulation in the first years, and a rapid stagnation in the years thereafter. I propose a subsidy scheme that takes into account the tendency of (potential) prosumers is to invest to maximum capacity. For any preferred PV-adoption level in 2030, there exists a corresponding ‘ideal’ subsidy path that ensures a smooth transition, at limited costs. I start out with a simple net-cost minimization model. I expand this to a calibrated consumer decision model that accurately describes the PV investment decision of potential prosumers. The minimization problem gives the optimal investment decision for the rational consumer (prosumer). I show that in the current regulatory framework, it is always optimal for the (rational) consumer to invest and become a prosumer. The prosumer will set the production capacity such that it exactly covers one’s own consumption. However, observed PV adoption levels range from 0.025% in 2010 to 8.8% in 20171. Therefore, the model is calibrated to these observed market conditions. The calibration parameter affects the (perceived) fixed costs of investment, and follows a log-normal distribution. The distribution parameters are estimated using 28 pairs of percentiles of actual PV adoption for the years 2010-2017. The calibrated model is used to provide an answer to the main research question: “What is the optimal model structure to study the investment decision of potential electricity prosumers, and what is the resulting optimal feed-in subsidy path to ensure a gradual growth path towards the natural, desired level of PV adoption for the years 2020-2030?” 1 For the group of interest in this thesis, which makes up about 47% of the total population.
I estimate the distribution parameters of the calibration parameter and use it to study and simulate the effect of a variety of different incentive schemes for the years 2010-2030. The estimated parameters of the log-normal distribution are 𝜇 = 2.5327 and 𝜎 = 0.4023. This indicates that the median consumer experiences ‘perceived’ fixed costs of 12.59 times the factual fixed costs of investing in PV2. The calibrated model is then used to predict a PV adoption level of between 90% and 99.78% in 2030 within the target group, depending on the type of incentive scheme and assumed model parameters. Besides the end level of PV adoption, the growth path is considered to be important, and should be as smooth as possible. Considering all this, I advocate a gradual subsidy path starting at €0.0890 per kWh in 2020 and decreasing to €0.017 in 2030, instead of the €0.159 in 2020 to €0,00 in 2028 dictated by the simple payback-period analysis advocated by the Dutch government (Wiebes, 2018). The predicted values depend heavily on the assumed parameters and inputs of the model. However, the inputs are well-founded, probable and reasonable. If more accurate data does exist, however, the model allows for even more accurate output. The model could therefore be used to set the desired subsidy path, according to the requirements set by the regulator and the most accurate data and inputs available. Recently, a letter was sent to parliament by the minister of economic affairs, Wiebes, outlining the future of multiple renewables-incentive schemes, including net-metering3. It is to be replaced, in 2020, by a feed-in subsidy for electricity sold to the grid. Although net-metering has been one of the driving factors of the rapid growth of solar PV adoption by households in the Netherlands, it is relatively costly compared to other incentive schemes aimed at larger renewable energy solutions. Furthermore, the expectation is that soon it will cause a high degree of overstimulation. Nevertheless, involving households and creating social acceptance is deemed an essential part of the energy transition in its entirety (Wiebes, 2018). The Dutch have committed to ambitious goals set in both the Paris Climate Agreement and the Dutch coalition agreement. In
2 The median is given by 𝑒m= 𝑒n.opnq= 12.59. By definition, 50% of the population experiences a lower value, and the other 50% a higher value. The log-normal distribution is right-skewed, as it can never be negative. In this instance, the log-normal distribution is assumed to model many (small) random effects that influence the prosumer’s propensity to adopt PV.
3 Net-metering allows prosumers to subtract the aggregate yearly PV production from the aggregate
order to achieve the 49% CO2 emissions reduction by 2030, agreed upon in the latter, incentives schemes remain necessary (Scarpa & Willis, 2010; Wiebes, 2018). On the other hand, to enhance social acceptance, a sustainable government budget policy is also key. Adoption of new technologies is often a gradual process, and to reach a high degree of social acceptance it is best to respect that natural tendency. Involving consumers is an essential part of achieving social acceptance for the energy transition (Wolsink, 2018; Wüstenhagen & Menichetti, 2012). A variety of studies show that solar PV will soon be one of the cheapest forms of electricity generation in many parts of the world (Mayer, Philipps, Hussein, Schlegl, & Senkpiel, 2015; Partain, et al., 2016). This points to the existence of a certain (high) natural level of PV adoption in the future. However, no microeconomic models exist that can accurately describe the investment- and adoption decision of consumers for photovoltaics. Although economists have attempted to model prosumer behavior (Sun, Beach, Cotterell, Wu, & Grijalva, 2013), no models exist to specifically describe prosumer investment behavior. Therefore, I introduce such a model based on cost minimizing behavior. However, in practice it shows that more aspects are important in the investment decision of potential prosumers. In order to incorporate these in the cost minimization problem, I introduce a ‘propensity to invest’ parameter that attempts to model the many factors at play in the investment decision. I assume a lognormal distribution of that parameter. I argue that this broader model is a better way to predict consumer behavior than the simple payback-period analysis that is now the preferred model by policymakers. The rest of this thesis is built up as follows: In section two, I outline the retail market for electricity in the Netherlands. This is the relevant market for the model, as prosumers act on the scale of individual consumers, but with some generation capacity. In section three, I briefly outline existing literature on the theoretical side of modelling prosumer behavior. In section four and and five, I introduce my model and its main inputs. Section six gives gives an application of the model for the years 2020-2030. Using simulations, I study the proposed policy implications. The results of this are discussed in section seven. Section seven also gives the shortcomings of the model and its results, and proposes possibilities for further extension of the model. An overview of the final conclusions is given in section 8. Lengthy mathematical derivations and large tables of output can be found in the appendices.
2 Dutch retail electricity market Electricity markets are different from traditional commodity markets. Goods like grain, sugar and even energy products like gas, coals and natural gas can be stored and traded at any moment. Electricity on the other hand, needs to be consumed immediately after being generated. Although increasingly available, storage is not yet an economically viable option for most purposes. Therefore, the electricity grid needs to be in-balance at any point in time. Traditionally, this market was characterized by only one or a few producers, and many individual consumers. As the energy transition develops, a third entity has made its entrance. Its size is comparable to the consumer, but it both consumes and produces electricity, mostly with solar panels on its own roof: the prosumer. Like with all commodities, different types of markets exist for electricity. Although the ‘physical product’ itself is completely homogenous, the differentiated sources from which the electricity originate allow the consumer to choose a heterogeneous product. Also, electricity is traded on spot markets as well as future markets. Both these spot and future markets are important instruments in balancing the grid. Future and spot markets both serve as a balancing mechanism. The price decreases when there is overcapacity, so generators are switched off. Vice versa, prices are higher at peak times, incentivizing producers to turn on generators and cover the excess demand. Because my model endeavors to accurately describe the consumer and prosumer markets, I only consider the retail market. Before 2004, electricity consumers were tied to the electricity monopolist of their region. From 2004 however, the electricity market was completely liberalized and consumers were free to choose a supplier (van Damme, 2005). Consumers generally choose a supplier that offers them a fixed-price contract for a certain duration, after which the consumer may choose to extend the contract or switch to another supplier. Retail consumers in the Netherlands are never active on the real-time electricity market. The energy supply company is the balance responsible party4 on their behalf. This supplier predicts the consumer’s use on the basis of its 4 Parties responsible for informing the grid operator of planned production, consumption and transportation needs, for the purpose of balancing the grid. For more information, see https://www.tennet.eu/electricity-market/dutch-market/balance-responsibility/
historical use (‘annual standard usage’) and a predicted consumption profile5. The energy supplier also sets the electricity prices, considering the (expected) spot- and future market prices, and the consumer chooses a supplier that best suits its preferences. Besides from the price, consumers decide on level of service, and origin of the electricity (green or grey). Although intense competition has driven electricity prices down and closer together in the past few decades, (perceived) heterogeneity and (perceived) switching- and searching costs allow for price dispersion still (Mulder & Willems, 2016). Consumers are sometimes afraid of switching, but as consumers get used to a liberalized market, the amount of switches is expected to increase. Energy supply companies may exploit consumer passiveness by offering different contracts for different types of consumers (price discriminate) (Hyland, Leahy, & Tol, 2013). This is demonstrated by the existence in the Norwegian retail market of a highly competitive segment with low margins, as well as a monopolistic segment with high margins (von der Fehr & Hansen, 2010). Currently the Dutch electricity market for consumers and prosumers is characterized by yearly settlement, with fixed-duration contracts and prices. Consumers pay a monthly advance payment that covers their expected expenses. At least once per year, the actual electricity meter readings are registered, and used to calculate the actual usage in order to settle the year’s energy bill. Consumers are free to choose and change an energy supply company whenever they like, given that they stick to their contract. Contracts are typically available from an indefinite duration (freely terminable) to one, three or even five years with fixed prices. Breaking the contract and switching early induces a fine. The retail energy market in the Netherlands is competitive since the liberalization of this market. It is characterized by low information- and switching costs. Many online price-comparison platforms are available, and due to strict guidelines switching is considered easy and free of charge. From a regulatory point of view, prices within the same contract for buying and feeding-in may differ, but supply companies generally set the same price for both6. 5 A consumption profile is the ‘spread’ of electricity consumption over the year. A profile fraction exists for every 15-minute interval of the year. The sum of profile fractions of one year equals 1, by definition. 6 The regulating authority, ACM, has only determined that the supply company should offer a ‘fair price’ for feeding in electricity. For more information about the ACM’s regulation of the retail energy market, see (Authority for Consumers & Markets (ACM), 2016)
For prosumers, the total amount of quantity bought from and sold to (fed in) the grid, which are registered and recorded separately on (modern) electricity meters, can be off-set once a year. This is a common tax-credit incentive scheme for renewable energy that is also known as net-metering. A prosumer only pays taxes over its yearly net consumption (electricity use minus self-generation). This is a generous scheme for a variety of reasons. Firstly, the grid remains an important utility for the prosumer, since its consumption and generation are not aligned. During the night and on cloudy (winter) days for instance, almost all the electricity consumption needs to be bought from the grid. On sunny summer days on the other hand, the overproduction is fed into the grid. Under net-metering, the total power sold to the grid is subtracted from the total quantity bought. Only the net consumption is subject to taxation. Variable taxes make up between 60-75% of the (variable) electricity price in the Netherlands. Initially devised to allow prosumers to consume their power generation whenever they like instead of when it was generated, it is currently the main driver of investment in PV-capacity on a household level (Wiebes, 2018). As a result of this and decreasing prices for photovoltaics, PV-adoption in the Netherlands has grown. However, net-metering also has downsides. It is essentially a generous subsidy scheme. This makes it costly for any government to allow net-metering. In terms of cost-effectiveness, it is more expensive than incentive schemes that support large-scale renewable electricity production (Wiebes, 2018). Furthermore, installing overcapacity is not attractive for prosumers, where doing so might actually be efficient and socially desirable. Prosumers only get the base price over their overproduction. Furthermore, consumers hardly have an incentive to reduce their own consumption once the generation capacity is installed, because that induces the same low marginal benefit. Also, prosumers have no incentive to align their consumption pattern with their own production pattern, as only the net-purchase of electricity over the year is considered for electricity taxation. Scarpa and Willis (2010) underline the importance of a regulator to step in, if significant growth in distributed generation capacity is to be achieved. However, the effects of net-metering are not longer aligned with the requirements of a desired incentive scheme. For these reasons, the regulator has announced the net-metering regulatory framework will be replaced by a feed-in subsidy scheme in 2020.
A number of alternatives to net-metering have been proposed, such as an investment subsidy or quantity-limited net-metering. In June 2018, the minister of economic affairs, Eric Wiebes, has announced the end of net-metering at the start of 2020. It will be replaced by a feed-in subsidy. This subsidy will likely be lower than the tax in order to be financially viable. Wiebes further announced that the feed-in subsidy will likely be calculated based on a simple payback-period analysis. The intention is to use the feed-in subsidy as an instrument to set this payback period to seven years, for as long as this is necessary7. In this new system, the full quantity bought from the grid is subject to taxation. In return, the prosumer is entitled to a certain subsidy per kWh fed into the grid. No limitation on the quantity eligible for subsidy is mentioned, but this will likely be considered. Furthermore, the Authority for Consumers & Markets (ACM) has started with pilots to allow multiple energy suppliers per connection (household). This means that in the (near) future, prosumers will be allowed to buy their non-self-generated electricity from a different supplier than the firm to which they sell their excess production. The table below displays the electricity bill breakdown for a representative prosumer under net-metering in 2018. Because of the net-metering regulatory framework, the prosumer benefits from two tax credits. Net-metering allows the prosumer to subtract quantity fed-in from the electricity bought. Only the net purchase is subject to taxation. As an added benefit, the prosumer is also entitled to the fixed tax credit. In case of a lower PV capacity, the electricity tax of €0,105 is induced over the net quantity purchased from the grid. If the PV capacity is higher, the excess quantity is only worth the base price €0,052. For the remainder of the analysis, only the variable costs are considered, as only those are important in determining optimal investment behavior. My definition of a prosumer in this paper is an electricity consuming household that has the capability to put up a PV-system on the own roof. It excludes PV-systems rented by housing corporations and systems installed by cooperations of consumers. 7 Further details of how this is to be determined were not shared
Description Quantity Price Amount (excl. VAT) Amount (incl. VAT) Variable costs Electricity bought 2256.3 kWh €0,052 €117,33 €141,97 Electricity sold -2256.3 kWh €0,052 -€117,33 -€141,97 Electricity tax 0 €0,105 €0 €0 Renewable energy markup 0 €0,012 €0 €0 Fixed costs Fixed fee 365 days €0,05 €18,25 €22,08 Network operator fee 365 days €0,5384 €196,52 €237,78 Fixed tax credit8 365 days -€0.8453 -€308,53 -€373,33 Total - - -€93.76 -€113.47 Table 1 - Electricity bill breakdown for the representative prosumer under net-metering 3 Literature review In economics, a wide range of models exist to study the economic behavior of both consumers and producers. In the field of microeconomics, consumers generally maximize utility subject to a budget constraint. Firms maximize profits, given the demand curve determined by the consumer’s preferences. Furthermore, investment decisions may play a role in optimal producer behavior. Economics usually treats consumers and producers as being mutually exclusive: an entity can’t be a producer and a consumer at the same time. However, that is exactly what households with solar panels are: energy prosumers. Potential prosumers want to maximize utility from energy consumption, but they also need to make an investment decision, after which they expect to gain a certain profit from their investment. Classic microeconomic models fail to describe such behavior. Since liberalization of retail energy markets has become more common, more economic literature has appeared that endeavors to study and model these markets. Ventosa et al. (2005) find a wide variety of ways to model behavior in energy markets, most of which involve optimization-, and simulation approaches. Furthermore, El-Khattam, Bhattacharya, Hegazy & Salama (2004) introduce a model to study the optimal investment levels for distributed energy generation. However, the model does so from 8 This is a tax credit that every household in the Netherlands with an electricity is entitled to. This is independent of total consumption. The variable electricity tax is quite high, to stimulate economization on electricity. The purpose of this tax credit is to compensate the average household for the high variable tax, while still incentivizing households to consume less electricity. The amount including VAT is paid out.
the perspective of utility companies. Producers and consumers are still two different entities, and they fail to appreciate the increasing importance of distributed prosumers. Although some recent literature focusses on the twenty-first century energy challenges (Pfenninger, Hawkes, & Keirstead, 2014), most of the models study optimal behavior in energy markets on a macro-scale. Consumer- and producer behavior may seem incompatible from an economic point of view. Some methods to harmonize the two have been proposed. The prosumer problem is analogous to the problem of owner-controlled firms. Such firms (owners) also have to amalgamate the objectives of firm profit maximization and personal utility maximization (i.e. maximize consumption). Other proposals include models of cost minimization, and equivalent expenditure models. For instance, Hubert & Grijalva (2011) propose a tool that allow prosumers to manage their consumption, generation and storage in order to minimize total expenditure. However, as most models that study prosumer behavior, they consider the generation capacity to be fixed. An attempt to model prosumer behavior from an economics point of view is undertaken by Sun et al. (2013). They combine traditional consumer and producer optimization models to devise a model capable of describing prosumer behavior. The model is helpful to understand the basics of the trade-offs at hand, but is very limited in its practical use due to some unrealistic assumptions. The main accomplishment of the paper is, arguably, the successful combination of consumer- and producer problems in a classical micro-framework. In this framework, consumers maximize utility subject to certain constraints, and producers maximize profits subject to market conditions. The maximized quantities are different and non-interchangeable. As a solution, Sun et al. (2013) modify the optimization problems in such a way that they become compatible. The producer problem is turned into a minimization problem by considering the negative profits. They combine this with the consumer’s dual (cost-minimization) problem subject to a minimum utility constraint, as well as a balance constraint that dictates the total inflow of electricity (grid purchase, generation) equals the outflow (grid sale, consumption). The combined problem is then transformed back into a utility maximization problem that can be solved using Lagrange. The traditional budget constraint is complemented by some revenue from the sale of power to the grid. The authors introduce a number of extensions that attempt to make the model more useful in real-world situations. It allows for different preferences towards buying or selling
power, for instance. They argue that this is analogous to preference for ‘green’ over ‘grey’. Sun et al. (2013) therefore assert that their model is useful for studying prosumer decisions regarding environmental objectives. Although useful as a baseline model and certainly an improvement to previous models, this formal economic model has its limitations. In the Sun et al. model (2013), the quantity of generated power 𝑄x is a constant that is not a choice variable. That assumption makes the (unaltered) model inadequate for studying the investment decision of the (prospective) prosumer. The model was not designed to study the investment decision, but considers the investment costs as sunk. The model theoretically allows for a difference between the selling and buying price of power, 𝑃z and 𝑃{. However, this poses a problem. If these prices differ the model allows
for (unlimited) arbitrage. It assumes that prosumers can buy power indefinitely from the grid and sell that same power back to the grid, resulting in high arbitrage profits. As a solution, Sun et al. (2013) allow for different preferences for buying and selling power, reflecting consumer tastes for ‘green’ versus ‘grey’. In reality, the quantities bought and sold (𝑄{ and 𝑄z) are not choice variables, but they are functions of the quantity
consumed (power load), the quantity generated and the distribution of these over time. It is unreasonable to assume that prosumers can loop electricity from the grid through their meter, gaining a profit. Furthermore, the authors assume that prosumers proactively respond to real-time price fluctuations on the electricity market. Given the structure of many retail energy markets, that is not a reasonable assumption9. And even if they could, it is questionable whether they would. This is reflected in studies that show short-term demand elasticities for electricity are rather low. Espey & Espey (2004) find a short-term demand elasticity of about -0.35. Depending on the setting, some studies find short-run demand elasticities of -0.012 (Labandeira, Labeaga, & López-Otero, 2017, p. 556). Estimates widely range depending on methods used and setting chosen. Jamil & Ahmad found that both price and income elasticities of demand are small and insignificant in the short run (2011). Furthermore, even in a setting of a big shock to energy prices such as in the 1973/1974 energy crisis, Bentzen & Engsted find no evidence of a structural break in energy demand as a result of this (1993). 9 In fact, it is prohibited and therefore impossible for anyone without an energy supply permit to act on the wholesale electricity market.
Considering this, it is not realistic to assume that consumers change their electricity consumption according to changing market conditions in the short term. For many consumers and prosumers, electricity is something that is necessary for the basic functioning of their homes and lives. They can’t be assumed to actively alter this at any given moment in order to ‘maximize their utility’. Consumer’s consumption patterns are relatively fixed in the short run. This setting makes it a more realistic assumption to state the prosumer problem in terms of a conscious investment decision in which prices and consumption are fixed for a certain duration, for instance a year. The consumer will minimize the costs of the electricity consumption by choosing between investing in PV capacity or not. There are other considerations in the decision whether or not to adopt PV capacity other than monetary motivations. This may be a ‘feel-good factor’ of contributing to the energy transition, and reduced dependence on the national energy grid. Such assumptions are supported by a variety of studies. Firstly, there is a range of studies that have found a positive willingness to pay for green, renewable energy over ‘grey’ alternatives (Sundt & Rehdanz, 2015; Borchers, Duke, & Parsons, 2007; Ma, et al., 2015). Because the direct quality or physical characteristics do not differ between ‘green’ and ‘grey’ energy, this positive willingness to pay can only stem from some sort of ‘good feeling’ or (shared) sense of responsibility. Furthermore, Scarpa & Willis (2010) find that there is a positive willingness to pay for owning (renewable) energy generation capacity. However, that value was, in 2010, not sufficiently large for most households to cover the (capital) costs of installing such capacity (Scarpa & Willis, 2010). An incentive scheme is therefore necessary to support the wide adoption of solar PV capacity.
4 The Model In this section, I introduce a model to describe the photovoltaics-investment decision of potential prosumers. Whether or not they will invest depends on a variety of variables, parameters and relations. First, I describe the dynamics with which the quantities bought and sold are determined. These are not choice variables, as they are determined by the capacity 𝐾 and the total annual quantity consumed 𝑄~. Furthermore, as consumption and generated are highly dispersed, it is the dynamics of consumption and generation that ultimately decide the quantities that need to be bought and sold. These dynamics are then used to formulate a formal, theoretical model. In this model, the potential prosumer sets the capacity K such that his total costs are minimized. The model is calibrated to actual observed market conditions in a later section. 4.1 Modelling market quantities Regulatory and infrastructural limitations and grid structure prohibit the prosumer from looping electricity through the meter to gain unlimited (arbitrage) profits. This intuitive certainty needs to be formalized for the model, however. The aggregate PV-generation is determined by the capacity 𝐾. Let 𝑞€x(𝐾) be the quantity generated in the n-th 15-minute interval of the year. This quantity is determined by the capacity K, as well as stochastic factors, mainly the weather. On the yearly scale however, the sum of production, 𝑄x(𝐾) , behaves rather deterministic. It turns out that the variance in annual production is small10. Therefore, for this model, assume that the prosumer knows the resulting annual electricity production with certainty. It is a function of K and some efficiency factor 𝜂 that determines how efficient the prosumer’s installation is11: 𝑄x = 𝜂𝐾 (1) Where 𝑄x is given in kilowatt hours (kWh) and 𝐾 is given in watt-peak (wP). 𝜂 ranges between 0.85 and 0.95 in the Netherlands12. 10 For more information on the expected annual generation and the efficiency factor 𝜂, see: (van Sark W. , 2014) (Dutch only). 11 This depends on a variety of factors, such as the quality of the system, but also the location and average sunshine 12 This will be outlined in the section ‘Data and model inputs’
Let the dynamics of power consumption and generation be described by vectors 𝜑~ and 𝜑x. I will refer to these as consumption and production profiles, respectively. These profiles are defined as follows: 𝜑~ = 𝑓€„…~ … 𝑓€„‡~ 𝑤𝑖𝑡ℎ 𝑓€ ~ ‡ €„… = 1 𝜑x = 𝑓€„… x … 𝑓€„‡x 𝑤𝑖𝑡ℎ 𝑓€x ‡ €„… = 1 (2) In which every element 𝑓€ is a profile fraction. The sum of these profile fractions within a profile is 1 by definition. A vector of profile fractions is the distribution of usage or production over time. It gives for every 15-minute interval (n) of the year, the fraction of the total annual consumption or generation. In reality, this vector is continuous. It is determined for every moment in time, since electricity meters register the quantities traded with the grid at any given moment. However, the smallest time unit in which it is generally measured, and also the time unit at which the electricity market is settled, is at 15-minute intervals13. Therefore, I specify the model at N=35040 (15-minute) intervals, spanning 1 year. 𝜑~ is determined by the consumer’s consumption pattern. 𝜑x can never be directly altered by the prosumer. It is determined by factors out of its control, mainly the weather (sunshine). In this model, both profiles are considered constant and identical for all consumers and prosumers. Let 𝑄~ be the power load, which is the annual electricity consumption of the prosumer. Utilizing the profiles and taking total power load and generation capacity into consideration, the vector for quantities traded with the grid for each 15-minute period of the year (𝑞{z) is derived: 𝑞{z = 𝑓€„…~ … 𝑓€„‡~ 𝑄~ − 𝑓€„…x … 𝑓€„‡x 𝑄x (3) 13 This time unit is often referred to as ‘Program Time Unit’ (PTU). For more information on grid balancing, see https://www.tennet.eu/our-key-tasks/security-of-supply/security-of-supply/
This vector gives the over- or underproduction of the household per 15-minute interval. These quantities have to be bought from or sold to the grid. This is a fully automatic process, so I assume for this model that these quantities can only be altered by the prosumer by setting K14. The net annual quantity bought from (or sold to) the grid is the sum of this vector: 𝑄{z = 𝑞{z€ ‡ €„… ≡ 𝑄{− 𝑄z = 𝑄~ − 𝑄x (4) This satisfies the (less binding) balance constraint from Sun et al. (2013) that dictates the total inflow of power should equal the outflow plus consumption: 𝑄{+ 𝑄x = 𝑄z+ 𝑄~ (5) For the current regulatory scheme, because we know that the sum of each individual profile equals 1 by definition, the net consumption (𝑄~ − 𝑄x) is enough to calculate the tax burden. Accounting happens once per year, and taxes are only paid over this net quantity. The quantities traded with the grid need not be known by the tax authority. The analysis is complicated when these quantities need to be accounted for individually, as under the proposed regulatory reform. The quantity bought (𝑄{) is calculated by taking the positive values within 𝑞{z and aggregating them. The quantity sold (𝑄z) is calculated by adding up the absolute value of all negative values of the vector 𝑞{z. In other words, for a whole year, the quantities bought and sold are calculated as follows (where 𝑞€~ indicates that period’s usage, and 𝑞 €x the period’s generation): 𝑄{ = 𝑞€~ − 𝑞€x 0 • €„… 𝑖𝑓 𝑞€~ > 𝑞€x 𝑖𝑓 𝑞€~ ≤ 𝑞 €x (6) 𝑄z = 𝑞€ x− 𝑞 €~ 0 • €„… 𝑖𝑓 𝑞€ ~ < 𝑞 €x 𝑖𝑓 𝑞€~ ≥ 𝑞€x (7) 14 Which means indirectly setting 𝑄x, as 𝑄 = 𝜂𝐾
These can’t simply be described by differentiable functions. However, if one knows or can accurately predict 𝑄~, 𝜑~, 𝜂𝐾 and 𝜑x as is the case generally, one can predict the yearly quantities sold and bought. This is important for the investment decision of prosumers under the new tax regime. It does complicate the analysis however, since the lack of an elegant formula impairs the capability to solve this problem using Lagrange. Therefore, simulations will be necessary to study the model’s implications. 4.2 Baseline theoretical model In this section, I introduce the cost-minimization problem of the consumer. The model is loosely based on the model introduced by Sun et al. (2013), but I have altered it to accommodate specific requirements imposed by the central research question of this thesis. The model can be specified at the 15-minute interval. However, given the conditions in the retail market, consumers are assumed to minimize their electricity spending for yearly periods. If, however, conditions change and the infrastructure is in place to allow consumers to act on the spot (15-minute) wholesale market, the model may be adapted to facilitate such a situation. Consumers generally minimize costs subject to certain constraints. For consumer decisions between two or more normal products, this is often a utility constraint. However, the trade-off in my model is different. The potential prosumer does not choose between two products it consumes. Rather, the consumption is considered constant and it chooses the best way to achieve that consumption of a single product (electricity). The consumer has to decide whether to buy its full consumption 𝑄~ from the grid against price 𝑃{, or to invest in PV-capacity. Investing in PV capacity will lower the amount of power that needs to be purchased from the grid, 𝑄{. These avoided electricity market purchases are an important part of the benefit of investing in PV, as the marginal benefit of this equals the total price, including taxes. Furthermore, the leftover production during certain peak-production times 𝑄z (daytime, summer) is sold to the grid for price 𝑃z. The buying- and selling price are determined by the market price for electricity 𝑃”, and the electricity tax 𝜏 and feed-in subsidy 𝜐, respectively. Thus, market prices are: 𝑃{= 𝑃”+ 𝜏 𝑃z = 𝑃” + 𝜐 (8)
Furthermore, investing in capacity 0 ≤ 𝐾 ≤ 𝐾 incurs some cost 𝐶˜. If the consumer decides to invest in PV and becomes a prosumer, it chooses K such that the total costs are minimized: min ˜ 𝑃{𝑄{− 𝑃z𝑄z+ 𝐶˜𝐾 𝒔𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐: 0 ≤ 𝐾 ≤ 𝐾 𝐾 =𝑄x 𝜂 (9) Where the quantities bought and sold are determined by the consumption and production profiles and aggregate quantities consumed and produced: 𝑄{ = 𝑓¡ 𝑄 {z = 𝑞€ ~ − 𝑞 €x 0 𝑖𝑓 𝑞€~ > 𝑞 €x 𝑖𝑓 𝑞€~ ≤ 𝑞 €x ‡ €„… 𝑄z = 𝑓¢ 𝑄 {z = 𝑞€ x− 𝑞 €~ 0 𝑖𝑓 𝑞€~ < 𝑞 €x 𝑖𝑓 𝑞€~ ≥ 𝑞 €x ‡ €„… (10) Where the vector of grid activity (purchase and sale) is: 𝑄{z = 𝜑~𝑄~ − 𝜑x𝑄x (11) If the consumer does not invest in PV, this problem collapses into the simple (constant) annual cost function 𝐶𝑜𝑠𝑡 (𝑛𝑜 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡) = 𝑄~𝑃{ (12) This situation can change for every year studied, as prices, taxes and investment costs generally change over the years. However, I have suppressed this time notation for readability. The model assumes constant consumption 𝑄~ and fixed production and consumption patterns (profiles). Also, prices are the same for every consumer and prosumer, given the year. The cost function 𝐶˜ is in monetary terms and is determined
by the capacity K. Once invested, the prosumer can not disinvest. The prosumer can however increase its capacity K if doing so is optimal. An effect of the erratic nature of the profiles is that the minimization problem can not be solved using conventional mathematical analysis. Rather, all possible investment decisions are evaluated, and the consumer chooses the optimal one. An overview of the symbols used in the model is given in table 2. A schematic display of the consumer- and prosumer optimization problem is given in figure 1. Symbol Explanation 𝒒𝒏𝒃; 𝒒𝒏𝒔; 𝒒𝒏𝒍; 𝒒𝒏𝒈 The per time period quantity of power bought, sold, load (usage) and generated, respectively 𝑸𝒃; 𝑸𝒔; 𝑸𝒍; 𝑸𝒈 Annual quantities of power bought, sold, load (consumption) and generated, respectively 𝑷𝒆; 𝑷𝒃; 𝑷𝒔 The market electricity (e) price excluding taxes / subsidies, and the ultimate buying and selling prices per period, respectively. 𝝉 Per-unit (𝑄{) electricity tax 𝝊 Per-unit (𝑄z) electricity feed-in subsidy 𝑪𝑲 Per-unit (𝐾) cost of investment. Given by the levelized cost of electricity (LCOE). This is a personal cost function for the individual prosumer. It consists of a common (monetary) part and a personal ‘perceived’ cost part, including factors that are not purely monetary. Only induced if the consumer decides to invest and become a prosumer. 𝜼 The efficiency of the PV-system. Ranges between 0.85-0.95 for modern (new) systems in the Netherlands, depending on many factors. Determined mostly by the weather conditions, but also by the system’s efficiency. Assumed to be constant and equal for all prosumers in the Netherlands. 𝑲 Installed capacity of solar panels. Treated as a near-continuous variable in the model15. Given in watt-peak (Wp). Actual production given in kWh depends on 𝜂 and involves the prosumer’s personal situation (roof orientation, location) and the system’s efficiency. 𝑲 Maximum to the prosumers’ capacity potential (i.e. roof size limitations) 𝝋𝒍 Distribution of usage over time. Vector of 𝑁 = 35040 time periods. 𝝋𝒈 Distribution of PV-generation over time. Vector of 𝑁 = 35040 time periods. Table 2 - Symbols used in the model 15 Although in reality K is discrete, recent technological advances allow for treating this variable as near-continuous. There is an increasing amount of producers of solar shingles, an example is the Tesla Power Roof: https://www.tesla.com/nl_NL/solarroof
Figure 1 – schematic display of the consumer- and prosumer optimization problem. Consumers first decide, in each period, whether or not to invest and become a prosumer. Once invested, they cannot disinvest. However, the prosumer can set a higher capacity if doing so is optimal. Figure 1 - Schematic display of the consumer/prosumer investment decision
5 Data and model inputs Before I can use the model to study optimal investment behavior I need to determine the inputs of the model. In this section, I describe the model data and inputs required to study the Dutch market. First, I introduce actual usage and generation profiles. Then, I determine the cost structure of investment (𝐶˜). This cost function is described using the levelized cost of electricity (LCOE). Next, I describe the past, current and future market prices for electricity. Ultimately, limitations to the maximum capacity 𝐾 will be discussed, as well as the current investment levels into photovoltaics. 5.1 Usage and generation profiles Approximations of actual usage profiles are publicly available through NEDU16. These are available for different types of usage: from small households to large industrial users, to a designated profile for street-lights. They are published for market parties to be able to accurately predict usage during any given 15-minute period. This is important in maintaining grid balance. Energy suppliers need to align the electricity consumption of their customers as accurately as possible, or they will incur additional costs (fines). PV-production profiles are not publicly available. However, these can be extracted from PV systems that have the capability to register the production for every 15-minute interval, spanning at least one year. To study the PV investment decision, the theoretical notion in the previous section is supplemented by real data. For the consumption profile, I have taken the profile coded as ‘E1B’, a common profile for households (NEDU, 2017). It is a vector of 35040 15-minute periods spanning the year 2017. The sum of the fractions equals 1 by definition. Obtaining a generation profile requires more work and data manipulation. I have obtained a vector of generation allocations17 from a private source within the Netherlands. The data is obtained through a balance responsible party (BRP)18 in the Netherlands. Because of privacy guidelines, it has to remain anonymous. It is the 2017 16 Vereniging Nederlandse EnergieDataUitwisseling: Dutch Organization for Energy Data Exchange. A platform for cooperation and data sharing in the Dutch energy market. 17 The per-period allocation is the amount of electricity (kWh) fed into the grid, registered by the balance responsible party (see below). 18 Parties responsible for informing the grid operator of planned production, consumption and transportation needs. For more information, see https://www.tennet.eu/electricity-market/dutch-market/balance-responsibility/
PV-production allocation vector of a medium-sized solar producer. Although somewhat larger than a typical prosumer, it is small enough not to have any scale advantages due to its size. The installation is situated on a roof and therefore its generation profile is comparable to a prosumer’s. Each period’s fraction is calculated by dividing the period’s production (‘allocation’) by the total yearly production. Some minor alterations were made to this data to fit the purposes of this study. Firstly, some negative allocations were measured, probably due to system-specific usage. This might have slightly contaminated the data. However, these negative allocations (usage) are small compared to the overall profile19. Therefore, I have set these negative volumes to zero for the purpose of this study. Furthermore, this PV-system’s capacity was upgraded halfway through the year by approximately 41.9%. Therefore, I manipulated the data by accordingly decreasing all the post-upgrade quantities. Furthermore, during the work to upgrade the system, it was disconnected for approximately 1.5 hours. I filled these 6 empty allocative slots with the data of 2 days later, when the weather and sunshine conditions were very similar (knmi.nl, 2017). These minor corrections are not expected to have a contaminating effect on the profile. Figure 2 displays three graphs. The first graph displays the usage profile, and the second the generation profile. The third graph is a combined electricity-market profile for a usage-covering PV system. It shows the quantities bought (positive) and sold (negative) for every 15-minute period of the year. 19 The largest negative allocation measured was -1 kWh. In comparison, the maximum was 127 kWh produced in a 15-minute period. The overall average is 14 kWh, including all nightly zeroes.
Figure 2 – The consumption and PV-production profiles used in this study. The first- and second graphs give the consumption and production profiles, respectively. The third graph gives the vector of grid activity 𝑞{z for a consumption-covering system size. Positive values indicate grid purchases, and negative values indicate sale to the grid. Summary statistics of the profile are given in appendix 1a. It is apparent that consumption and production are not aligned. Consumption in summer is low, and high in winter. This is the opposite of the production profile. Furthermore, PV-production during the night is always zero. To further illustrate, consider the following graphs in figure 3 displaying both usage and production profiles during a random (sunny) winter day and a random (sunny) summer day:
Figure 3 – Zoomed in view of the production and consumption profiles. All four figures give the profile of a single day (96 x 15-minute periods). The top two figures give a typical winter and summer day consumption profile, respectively. The bottom figures represent the winter- and summer day PV-production profiles, respectively. It is by these dynamics that the quantities sold and bought are determined. Without the use of a battery, shortage or excess production has to be bought from or sold to the grid directly, as (internal) balance is imposed by the laws of physics and grid structure. Considering the profiles and the quantities consumed and generated: what are the dynamics of the quantities bought and sold? Consider again equation (4) that gives the vector 𝑄{z . Taking the sum of the positive and negative values respectively, 𝑄{ and 𝑄z are easily calculated for any combination of 𝑄~ and 𝑄x 𝐾 . Assume a constant power load of 𝑄~ = 3500. This is the Dutch household average (energiesite.nl, 2018). The
following graph shows the dynamics of the quantities bought and sold for 0 ≤ 𝑄x 𝐾 ≤ 7000.
Figure 4 – Dynamics of the quantities bought and sold. Constant power load 𝑄~ = 3500. The x-axis displays system size options ranging from 0 ≤ 𝑄x 𝐾 ≤ 7000. The y-axis gives the quantities bought and sold. The increasing line gives the quantity sold, the decreasing line the quantity bought. The quantities bought and sold intersect at 𝑄~ = 𝑄x = 3500, as is dictated by the original balance constraint. Furthermore, because of the relatively high base load (minimum of usage), the quantity sold only starts increasing (slowly) at 𝑄x(𝐾) = 359, and it doesn’t rise above 100 kWh before 𝑄x(𝐾) = 834. Most of the generation at these system sizes directly decrease the prosumer’s quantity bought. These ‘avoided electricity market purchases’ are an important part of the revenue of a PV-system (Gil & Joos, 2008). They will gain importance under a new taxation scheme. As the quantity bought is nearing its minimum20 we see that less is consumed directly by the prosumer and more needs to be sold off to the grid. I will explore briefly the regulatory playing field that is created by these dynamics. Consider the common situation where 𝑄~ = 𝑄x = 3500. The quantity bought and sold will then also be equal: 𝑄{= 𝑄z = 2256.3 𝑘𝑊ℎ. Under net-metering, it would be 20 The overall minimum is 1804.1 kWh for very large system sizes. However, the minimum is about 2000 kWh for any realistic setting, considering the maximum to capacity K dictated by for instance the limited roof size.
enough to know the net quantity bought (or consumed), which is zero. The (variable) tax burden in that case is zero. More generally: 𝑂𝑙𝑑 𝑔𝑟𝑖𝑑 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑖𝑡𝑦 𝑐𝑜𝑠𝑡 (𝐶Á~ÂxÃÄÂ) = 𝑄{− 𝑄z 𝑃”+ 𝜏 −𝑄z𝑃” 𝑖𝑓 𝑄{ ≥ 𝑄z 𝑖𝑓 𝑄{ < 𝑄z (13)
Where 𝑄{− 𝑄z is equivalent to 𝑄~ − 𝑄x. Therefore, as long as 𝑄~ ≥ 𝑄x, one doesn’t need
the usage- and generation profiles in order to be able to calculate the market cost and/or tax burden. In this (net-metering) example, it is easily demonstrated that 𝐶Á~ÂxÃÄ = 0 (14) In a new system with separate tax rates and feed-in subsidies, the quantities bought and sold will both be important in determining the tax burden, not just their net value. In the new regulatory framework, where 𝜏 and 𝜐 are expected to differ, the cost function is: 𝑁𝑒𝑤 𝑔𝑟𝑖𝑑 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑖𝑡𝑦 𝑐𝑜𝑠𝑡 (𝐶€”ÅxÃÄÂ) = 𝑄{ 𝑃”+ 𝜏 − 𝑄z 𝑃”+ 𝜐 (15) And in this particular example, since the base price is the same for buying and selling, and 𝑄{ = 𝑄z : 𝐶€”Å = 𝑇𝑎𝑥 𝑏𝑢𝑟𝑑𝑒𝑛 = 2256.3(𝜏 − 𝜐) (16) More generally, the market electricity cost (grid expenditure) and tax burden (or revenue, if < 0) are given by: 𝐶€”Å = 𝑔𝑟𝑖𝑑 𝑒𝑥𝑝𝑒𝑛𝑑𝑖𝑡𝑢𝑟𝑒 + 𝑡𝑎𝑥 𝑏𝑢𝑟𝑑𝑒𝑛 = 𝑄{xÃÄ − 𝑄z 𝑃” ”ÌÍ”€ÂÄÎÏÔ + 𝑄{𝜏 − 𝑄z𝜐 ÎÐÌ {ÏÔ€ (17) Net-metering provides a high subsidy on generation, but only up to the point where consumption equals generation. It can’t be tweaked according to the needs and wishes of the regulator and society as a whole. Intuitively, one can derive that the new taxation scheme will create incentives for the prosumer to decrease and align its consumption towards a more socially optimal situation. It now becomes more rewarding to consume
one’s generated electricity at the time it is generated, relieving the grid. As storage capacity will increase and costs will decrease, it might even be beneficial to balance one’s own ‘micro grid’, potentially completely removing the necessity for a central balancing mechanism. Furthermore, prosumers might have more incentives to install excess capacity, as the marginal benefit is higher than under net-metering. This excess production can then be used by neighbors that don’t have a suitable roof, or by cities, which generally have less suitable space for PV-generation. Moreover, the increased marginal benefit incentivizes prosumers to decrease their usage, analogous to installing excess capacity. This effect is analogous to prosumers installing overcapacity, and only the latter is part of the model, since consumption is assumed to be constant. 5.2 Cost of generation: levelized cost of electricity (LCOE) Ample research has been conducted into the area of photovoltaic (PV) energy systems and their price developments. For (potential) prosumers, many factors are important in determining the total costs and benefit of installing a PV system. The total price of the system is the most important. Quality is another important factor, as a longer lifespan allows the prosumer to gain more revenue from the system and spread the investment over a longer period. A better quality is also expected to result in lower operating and maintenance costs. Furthermore, the interest rate plays a crucial role in the investment costs. As I will show in this section, prices have gone down considerably over time, while quality has steadily improved. What’s more is that the price of a system tends to decrease (relatively) as its size increases. All these factors have to be accounted for when modelling the investment decision of (potential) prosumers. Almost any study into the cost of PV make use of the same measure: the levelized cost of electricity (LCOE). The LCOE provides a simple method for comparing the per-unit cost of electricity generated by different sources. It can be used for instance to compare the cost of generating PV-electricity to any other technology, either fossil fuelled or any other renewable source (Short, Packey, & Holt, 1995). Campbell (2017) has made an important contribution in understanding and determining the factors that drive the costs of any PV system. In recent years, substantial progress has been achieved in driving down all factors that determine the total cost of PV systems (Campbell, 2017). The economics and pricing of PV systems are fundamentally different than that of traditional energy systems, like the fossil-fuelled ones. PV-systems have a higher initial
capital investment, but in return have lower operating- maintenance and other variable costs. It doesn’t require any fuel to work, other than sunshine (Campbell, 2017). The levelized cost of electricity (LCOE) is the most widely used metric for calculating the average cost per unit of electricity generated, something that the model in this paper requires (Campbell, 2017, p. 624). It allows one to compare different technologies for electricity generation. It was originally designed to be able to compare larger-scale, centralized generators of electricity, as is common with fossil-fuelled energy (Short, Packey, & Holt, 1995). However, it can be easily modified to facilitate requirements of my model. In short, the LCOE is simply the net present value of the total investment made, divided by the total output in terms of electricity production (Campbell, 2017, p. 624): 𝐿𝐶𝑂𝐸 = 𝑇𝑜𝑡𝑎𝑙 𝑙𝑖𝑓𝑒𝑡𝑖𝑚𝑒 𝑐𝑜𝑠𝑡 𝑇𝑜𝑡𝑎𝑙 𝑙𝑖𝑓𝑒𝑡𝑖𝑚𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 (18) Campbell (2017) provides an extensive formula for the exact LCOE intended for large-scale producers. In this paper however we will use the following form, that is introduced by van Sark & Schoen (2016, p. 9) and uses annual numbers: 𝐿𝐶𝑂𝐸 =𝛼𝐼 + 𝑂𝑀 𝑄x (19) And modified for my model: 𝐿𝐶𝑂𝐸 =(𝛼 + 𝛾)𝐼 𝜂𝐾 𝑤𝑖𝑡ℎ 𝛼 = 𝑟 1 − (1 + 𝑟)¢× (20) With the following symbol explanations, given in table 3:
Symbol Explanation 𝜶 Annuity factor 𝜸 Operating & maintenance cost (percentage of 𝐼) 𝑰 Initial investment sum 𝒓 Cost of capital / interest rate 𝑳 System lifespan Table 3 - LCOE symbols and explanations There are five main factors that determine the LCOE, all of which will be discussed separately: cost of capital, system lifespan and residual value, energy production, operating and maintenance and the initial investment made (total cost of the system) (Campbell, 2017). § Cost of capital (𝒓) In determining the cost of capital, there are multiple factors that play a role. The most important one is risk. There are many forms of risk, and many of the relevant risk factors are related to the prosumer’s personal situation. However, the risk of investment in PV itself is limited. The standard deviation of PV-output is very low, and the annual output rather predictable (Campbell, 2017). The quality of PV systems has gone up and this is reflected by decreasing interest rates for PV-system loans. In the U.S., as Campbell finds, rates decreased in only a few years from over 10% to less than 6% (2017). Perhaps one of the highest risks in the Dutch market is the insecurity about future regulation and taxation, as net-metering will almost surely disappear and potential prosumers don’t know what will replace it. However, subsidized loans for PV investment are offered by government institutions that offer interest rates as low as 1.6% for certain investments in sustainability, including PV (SVN, 2018). Taking this into consideration, cost of capital varies somewhere between 1.6% and 5%. For the model application, I assume an average constant interest rate of 3%. § System lifespan (𝑳) The lifespan of a PV-system is not a strict measure after which the systems stops producing. It should rather be regarded in terms of opportunity costs. Chances are that in 25 to 35 years’ time, technology has improved so far that the opportunity cost of leaving the old system on the scarce roof space becomes too high and it needs to be replaced. Although PV systems could probably operate for over 40 years with proper maintenance (Campbell, 2017). Most PV system are amortized over a period of 25 years
(van Sark & Schoen, 2016).Therefore, in the application I will assume a lifespan of 25 years. § Energy production (𝑸𝒈) Electricity production of PV-systems21 differs greatly around the world. It may be as high as 2.5 kWh/Wp in an ‘ideal’ place like the Atacama Desert in Chili22, but in Western Europe it is generally less than 1 kWh/Wp (Campbell, 2017). Most sources mention a per-Wp production in the Netherlands of about 0.85-0.95, depending on the location and on-site situation like roof direction, shade etc. (van Sark & Schoen, 2016; Londo, Matton, Usmani, Klaveren, & Tigchelaar, 2017). I assume the efficiency parameter is a constant in my model application. Therefore, the quantity generated is a direct function of K: 𝑄x = 𝜂𝐾. I assume an efficiency parameter of 𝜂 = 0.9 in the application of the model. § Operating and maintenance (𝜸) The operating and maintenance (O&M) costs for PV-systems have always been low compared to other technologies (Campbell, 2017). PV systems require no fuel and have almost no moving parts that need to be replaced. However, some O&M costs remain that become a more important factor as the other cost-determining factors go down steadily. For PV systems, annual O&M costs are generally expressed as a percentage of the initial investment sum. In order to keep the system as efficient as possible, regular cleaning and maintenance to surroundings is necessary23. Insurance is another cost that would fall under O&M costs. Additionally, inverters generally have a shorter lifespan than the solar panels. Replacing the inverter is also covered under the O&M costs. Commonly, O&M costs generally range around 1% or 1.5% (van Sark & Schoen, 2016; Londo, Matton, Usmani, Klaveren, & Tigchelaar, 2017). For my model application, I assume constant operating and maintenance costs of 1% of the initial investment sum. 21 Production/capacity ratio is often expressed as kWh/Wp per year, where Wp is the installed capacity (K), and kWh is the amount of generated output. 22 Because of its high altitude and clear skies, this desert is one of the most efficient places in the world for solar PV generation (Campbell, 2017) 23 For instance: cutting trees to remove shade, cleaning the PV system and removing other obstacles all needed for the proper functioning of the system are considered O&M costs.
§ Initial investment: cost price of PV-systems (𝑰) The initial investment cost of a PV-system is perhaps the most widely studied and definitely the most important aspect of the cost of PV, as it influences everything else. It is this price that ultimately the consumer decides to pay or not. Van Sark & Schoen (2016) have undertaken an extensive analysis of all the relevant costs in the Dutch PV market. They study the cost development on the basis of eleven assessments over the years 2011-2016. Their investigation considers the cost of all aspects of PV-systems. I use their findings on the cost of complete PV-systems. Figure 5 displays two graphs that are taken from this paper. They show all observations of complete systems and have system size on the x-axis and system price (€/kWp) on the y-axis. Figure 5 - PV-system price as function of system size. Taken from: (van Sark & Schoen, 2016, p. 24) We see that for tilted roofs, prices are slightly lower but the shape of the function is similar. The authors have also provided an overview of average system price for some common system sizes: System size (kWp) Price (€) Installation (€) Total price (€) System price (€) 0,6 1,69 0,60 2,29 1.374 2,5 1,46 0,40 1,86 4.650 5 1,30 0,30 1,60 8.000 50 1,17 0,20 1,27 63.500 500 1,00 0,15 1,15 575.000 Table 4 - System price for several system sizes. Taken from: (van Sark & Schoen, 2016, p. 31)
For the prosumers in this analysis, a 500 kWp system is not viable. With such system sizes, the cost dynamics will be different. Furthermore, van Sark & Schoen (2016) mention that on tilted roofs, most systems have a maximum of about 5 kWp with some outliers around 25 kWp. Therefore, I will not consider the 500 kWp system in determining the cost function. The 50 kWp systems, although somewhat sizeable for a regular prosumer, are considered to also accommodate a reliable cost function for larger prosumers that have larger available (roof) space on for instance stables or large houses. Looking at figure (5) one notices an (exponential) downward trend of the average total cost (ATC) function. Total cost functions that follow such a shape are often given in the following (quadratic) form: 𝑆𝑦𝑠𝑡𝑒𝑚 𝑐𝑜𝑠𝑡 𝐼 = Ϝ + 𝛽…𝐾 + 𝛽n𝐾n (21) Where 𝑊𝑝 is the capacity of the installation and ‘fixed costs’ Ϝ are only incurred if the investment is made. The cost function that best fits the data for systems in table (4), excluding the largest is the following: 𝐼 = 598.1844 + 1.531197𝐾 − 5.463438 ∗ 10¢á 𝐾n (22) For system sizes up to 5 kWp, the following linear approximation is acceptably accurate: 𝐼 = 631 + 1.5𝐾 (23) If I include the 50 kWp system into the linear approximation, the constant becomes quite large, which causes significant overestimation of small system costs: 𝐼 = 1295.50 + 1.245𝐾 (24) The linear approximation is less accurate when system size increases. These are estimates based on average system prices from a large dataset (van Sark & Schoen, 2016). The following table shows the average figures by van Sark & Schoen (2016) and
the parameters I estimated to fit them24. The quadratic cost function seems to perform best at every system size. Therefore, I will consider the cost function given in equation (22). System size (𝒌𝑾𝒑) Van Sark & Schoen (2016) Quadratic fit 𝟎. 𝟔 ≤ 𝒌𝑾𝒑 ≤ 𝟓𝟎 Linear fit 𝟎, 𝟔 ≤ 𝒌𝑾𝒑 ≤ 𝟓 Linear fit 𝟎, 𝟔 ≤ 𝒌𝑾𝒑 ≤ 𝟓𝟎 𝟎. 𝟔 €1374 €1514,94 €1531 €2042,5 𝟐. 𝟓 €4650 €4392,03 €4.381 €4408 𝟓 €8000 €8117,59 €8131 €7.520,5 𝟓𝟎 €63.500 €63500,53 €75.631 €63.545,5 Table 5 - Cost function fitted values and actual observed values (van Sark & Schoen, 2016) § Future cost of PV With most new technologies, a downward trend in prices is observed as the technology proliferates. A visual representation of this downward trend is often referred to as a learning curve. It is evident that costs tend to decrease as researchers and manufactures get more experienced, especially with relatively new techniques (Partain, et al., 2016). Sometimes referred to as ‘Swanson’s Law’, the learning curve for PV broadly indicates a 20% price decrease for every doubling of worldwide PV-capacity installed. And indeed, this ‘rule’ has held (with ups and downs) since approximately 1980 (van Sark & Schoen, 2016). As the proliferation of solar PV picks up speed, costs are also rapidly decreasing. In Germany, the feed-in tariff paid by the government for large-scale solar systems fell from about €0.40 in 2005 to €0.09 in 2014 (-77.5%) (Mayer, Philipps, Hussein, Schlegl, & Senkpiel, 2015). These feed-in tariffs are often a good indicator of the cost of PV, as they are meant to compensate the investor for the higher cost of this (relatively) new technology. Despite long being the most expensive form of electricity, the authors now expect that solar energy will soon be the cheapest form of energy in many parts of the world (Mayer, Philipps, Hussein, Schlegl, & Senkpiel, 2015). Considering different scenarios, ranging from pessimistic to optimistic, the authors further estimate that in Northwestern Europe, costs will drop to about €0.04-€0.06 per kWh in 2025 and to €0.02-€0.04 in 2050. This indicates a drop of one third in the period 2015-2025 (Mayer, Philipps, Hussein, Schlegl, & Senkpiel, 2015). In order to model this for the purpose of this study, consider the following cost price development: 24 Because these are already average (fitted) parameters, providing regression tables is not particularly insightful. However, this model can accommodate any alternative cost function if there is more precise data available for the relevant case study. The estimated cost function is deemed likely, given the extensive nature of the paper they are taken from (van Sark & Schoen, 2016).
𝐼Î 𝐾 = 𝜓Î𝐼
ê (25)
Where 𝐼ê is the cost function given in equation (22). An estimation of 𝜓 = 0.9647 suits the data in that study quite well as this gives an estimate of 𝐼΄…ê= 0.6981𝐼ê and 𝐼΄po= 0.2843𝐼ê. In another extensive study into the Dutch PV and the net-metering regulatory framework, the authors have given investment cost estimates for the years 2010-2030 for both small (2.25 kWp) and larger (4.5 kWp) system sizes (Londo, Matton, Usmani, Klaveren, & Tigchelaar, 2017, p. 38). Imposing the structure of equation (25) on these estimates yields 𝜓 = 0.9699 for both smaller and larger systems25. This order of magnitude seems indeed rather robust, given the findings of van Sark & Schoen of a price decrease from 2012-2016 of 15.2% (2016), this would indicate an estimate of 𝜓 = 0.9596. Taking everything into consideration, I assume 𝜓 = 0.963 in the rest of the analysis. 5.3 Market prices In this section I outline the average prices, taxes and (expected) feed-in subsidies for the retail market. The electricity tax of current and past years are available through the Dutch tax authority (Belastingdienst, 2018). These numbers are for households that consume (buy) less than 10.000 kWh per year, which is well within my definitions of consumers and prosumers. Retail market prices have troughed in 2017 and are expected to increase gradually over the next 12 years. The expected price increase will be substantial compared to current price levels, although somewhat lower than expected earlier (Centraal Bureau voor de Statistiek, 2017). The increasing price is the result of abandoning high overcapacities in Northwestern Europe (Centraal Bureau voor de Statistiek, 2017). This is only partially compensated by a more rapid capacity growth of renewables. For my model implementation I use the expected price developments as determined by Londo et al. (2017). The authors also make a prediction of the future electricity taxes. These expected taxes serve as a baseline level in my model. The minister of economic affairs and climate, Eric Wiebes, has indicated recently that in 2020 the net-metering guidelines will be replaced by a feed-in subsidy. Net- 25 See appendix 4a for the derivation and output
