THE IMPACT OF TARIFF REGULATION ON SMART GRID INVESTMENT

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THE IMPACT OF TARIFF

REGULATION ON SMART GRID

INVESTMENT

A model with demand uncertainty

Author: Daniël Schans Student number: S2469308 E-mail: d.schans@student.rug.nl Thesis supervisor: Machiel Mulder MSc Finance

Faculty of Economics and Business, Rijksuniversiteit Groningen Date: 07-06-2018

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Abstract

As smart grids become more prominent in today’s electricity system, it becomes increasingly important to address the investment opportunities associated. Given the significant investments required, it is important to determine which type of tariff regulation fosters smart grid investment. This thesis has sought to explore the benefits of investing in the smart grid from the viewpoint of a DSO and develop a model accordingly. The findings from the model suggest that price cap regulation offers greater incentives when investment costs are lower than the avoided investment in capacity enhancement, and yardstick regulation incentivizes DSOs more to invest when investment costs are higher than the avoided investment. The results are similar when an increase in cost reduction is included.

1. Introduction

The power system is undergoing rapid change. Since climate change is recognized as a real concern and significant future threat, the focus of policy makers, regulators and business executives has shifted towards the usage of alternative and low emission energy technologies for electricity production. An example of this increased interest is the UN Climate Change Conference that was held in Paris. This conference negotiated a global agreement on the reduction of CO2 emissions, among which increasing the use of renewables in the energy supply chain is a pillar. This implies a reassessment of the current power system in order to guarantee a stable, efficient and sustainable energy supply in the future (Zahedi, 2011).

In recent years, novel sustainable electricity generating techniques are being implemented. Consumers are becoming ‘prosumers’ and start taking control of their electricity bills by installing solar panels and increasing the efficiency of their energy use. On the production side, large wind parks and solar farms are increasingly being developed. Consequently, the supply of electricity is becoming more intermittent. Electricity networks have to be adapted in order to facilitate increased distributed generation (DG) and higher penetration of renewable electricity sources (RES) (Peças Lopes et al., 2007). Here, the passively managed distribution network is no longer sufficient. Instead, a shift towards a smarter, more communicative distribution network will occur. This also changes the role of distribution system operators (DSOs) significantly (CEER, 2015). Even though the electricity market is characterized by a liberalization wave the last decade, DSOs still operate in regulated environments. Regulatory mechanisms might spur, but also seriously hamper investment in the electricity grid. DSOs are expected to carry the main investment burden and since they operate in a regulated environment, it is important to study the relationship between electricity grid investment and regulation (Cambini et al., 2016). Therefore, this thesis takes the viewpoint of the DSO in determining the benefits of smart grid investment and developing an investment model.

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effect of incentive-based regulation on investments in networks (Guthrie, 2006; Joskow, 2008). However, still some authors argue that cost-based investments provide a more stable environment for uncertain investments in networks (Kahn et al., 1999).

This thesis contributes to the literature by determining the investment timing in the smart grid under different regulatory mechanisms. Investment timing is expressed as a threshold for the demand level. In order to do so, the real options approach is used (Dixit & Pyndick, 1994). Traditionally, DCF calculations are utilized for investment decisions, but this method lacks the ability to include mangerial responses to sources of uncertainty. Real options valuation does provide the possibility to include this flexibility in an investment decision. In this specific case, electricity demand is the uncertain factor. Here, it might be optimal to wait until demand is at a certain level before investing in the smart grid. This problem is analyzed using dynamic programming in order to determine the first moment to invest in smart technologies. Here, a distinction is made between three regulatory mechanisms.

The remainder of this thesis is organized as follows. Section 2 presents an overview of the current literature on this topic where two streams of literature will be discussed. Section 3 provides information on the electricity grid and section 4 explains the role of regulation. Section 5 further clarifies the method and model utilized and in section 6 the models will be tested and some inferences will be drawn. These will be discussed in section 7, after which this thesis will be concluded.

2. Literature review

Previous studies

The effect of regulation on investments decisions has received substantial attention in recent years. In their seminal research, Averch & Johnson (1962) found that typical rate of return regulation does not provide a firm with incentives to improve efficiency. Also, they stated that rate of return regulation is likely to lead to overinvestment, because firms know their allowed rate of return beforehand. This paper has provided the cornerstone for to a shift from rate of return (or cost-based) regulation to incentive regulation in the electricity distribution sector (Joskow, 2014).

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price cap regulation leads to more efficient replacement decisions compared to rate of return regulation. Guthrie (2006) surveyed the relationship between tariff regulation and investment with a focus on infrastructure industries. He states that the investment flexibility firms enjoy greatly alters the investment choices firms make. Furthermore, since firms cannot pass on their costs under incentive regulation, the author claims it discourages investment.

There is no consensus in the literature on the effect of regulatory mechanisms and investment in infrastructure innovation like smart technologies. Littlechild (2006) argues that incentive regulation has a positive impact on stimulating innovation, because it provides a suitable framework for cost reduction and improved efficiency. However, as Kahn et al. (1999) point out cost-based regulation provides more stability which will promote investments in uncertain industries. Empirical evidence on the relationship between regulation and investment in limited and in particular focused on other infrastructure sectors. Cambini & Rondi (2010) investigated the relationship between investment and regulatory mechanisms for a sample of EU energy utilities. Their results show that incentive regulation leads to higher investment rates than rate of return regulation. It must however be noted that the authors focus on a period between 1997-2007, which overlaps with the period incentive regulation started gaining more attention. Ai & Sappington (2002) found that the modernization of the U.S. telecom networks was accelerated by price cap regulation. However, they did not find significant evidence for differences in profit and revenue between price cap regulation and rate of return regulation.

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a binomial lattice they identified the optimal first time for utilities to start investing in the smart grid. All in all the research on regulatory mechanisms that foster investments in the smart grid is fairly limited.

3. The electricity grid

The conventional electricity grid

In order to grasp a better understanding of the problem discussed in this thesis, the way in which the electricity market is structured will first be discussed. In most Northwest European countries, the electrical power system was built at the beginning of the 20th century. Even though the system increased in capacity tremendously, the structure of energy transmission and distribution has remained largely similar. In the conventional power system, large scale production plants inject large amounts of electricity into the transmission grid. This electricity is then transferred to lower voltages and via distribution networks transported to end-users. The existing centralized electricity paradigm was driven by several factors. Economies of scale drastically decreased average costs of electricity production, which implied production facilities with more capacity were built. Moreover, utilizing alternating current instead of direct current enabled electricity networks to cover way larger distances with a significant reduction in energy losses. These large production sites were connected to the electricity grid, which meant resources were pooled. This enabled network users to reduce reliance on a single production site as other sites were able to compensate for a loss (US Department of Energy, 2007).

One of the fundamental technical properties of the grid is that supply and demand of electricity must equal one another, otherwise quality of supply will decrease severely. A staggering 90% of all power blackouts and instabilities originate in the distribution network. Obviously, this percentage needs to be diminished tremendously. These power blackouts might actually induce domino effects leading to isolation of certain regions, because of the hierarchical structure of the current electricity system (Farhangi, 2010). Although issues in the distribution network can be easily solved, operators want to avoid it.

Developments electricity system

In recent years, two developments have led to a call for a renewed and modernized grid. The first development in the power system architecture is the increased share of distributed generation (DG). Second, the increased usage of renewable power plants leads to a more intermittent supply of electricity, which the grid is currently not designed for.

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generation capacity going from less than a kW to tens of MW. Chambers (2001) limits the capacity of a distributed electricity generator to 30kW. However, in the literature no consensus exists concerning the upper limit of the capacity. Ackermann et al. (2001) found that the limit ranges from 1 MW to more than 100 MW. This implies that the definition of distributed electricity encompasses a broad concept. Ackermann et al. (2001) therefore proposed a definition applicable to a large majority of distributed generators. Connection to a lower/medium voltage network and proximity to households is crucial here.

Distributed generation is driven by the need to limit green house gas (GHG) emissions (Peças Lopes et al., 2007). This is done by increasing the use of renewables and combined heat and power (CHP) systems. Examples of these renewables include solar panels and heat pumps. Also, electric vehicles as potential distributed energy resource are widely regarded. However, connecting electric vehicles to the distribution network and recharging the electric vehicles without any control may lead to serious load problems during load hours (Hu, et al., 2015). Therefore, the power grid needs control mechanisms in order to prevent power outages.

The second development in the energy market is the increased usage of renewable based and larger power plants. The increased penetration of renewable energy technologies (RET) drastically alters the way in which the electricity network is planned, constructed and operated. Due to the irregular generating character of RET, injection of electricity into the grid is becoming more volatile (Veldman et al., 2010). Higher volatilty implies higher peaks of electricity demand, which the current electricity grid might not be able to deal with.

Several ways exist for tackling the developments noted previously. First, the grid can be extended. This allows the grid to facilitate both peak load and peak supply, because of the increased capacity (Mulder, 2016). Second, the grid can be made smarter. This basically implies that grid users generate more accurate, real-time information about network usage and accommodate their behavior to this information. Innovative information technologies have a major role in this and will make the grid more efficient (Colak et al., 2015). Smarter grids are associated with the replacement or upgrading of existing infrastructure (e.g. new sensors) and the application of ICT infrastructure in the distribution system. Because a significant part of the distribution capacity is only utilized to meet peak demand for very short time periods (Farhangi, 2010), it might be valuable to invest in technologies that optimize demand and supply.

Distribution system operators

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investments will be allocated to developing the distribution system (Eurelectric, 2013). According to Cambini et al. (2016), distribution system operators (DSOs) carry the main investment burden when it comes to investing in the power grid. DSOs have two responsibilities in particular. First, they are tasked with the efficient, safe, and secure transportation of electricity. Second, they are required to create and maintain the electricity networks (ACM, 2017).

Even though the electricity market has been subject to significant liberalization, DSOs are still regulated entities. Therefore, revenues are determined by regulators and DSOs need to balance the expected benefits of grid investments with the capital costs associated. In order to set up a framework that spurs grid investment, regulation must be regarded (Guthrie, 2006). In section 4, the role of regulation will be discussed in detail. Costs for DSOs are divided into operating (𝐶𝑜𝑝𝑒𝑥) and capital expenditures. Operating expenditures include, but are not limited to, operating & maintenance (O&M) costs and electricity costs. Capital expenditures are related to the acquisition or upgrade of the regulatory asset base (𝑅𝐴𝐵) of a DSO, like the required return on investment (𝑟) and depreciation 𝐷𝑡 (Guthrie, 2006). The costs that

influence regulatory settings can be calculated utilizing the following equation:

𝐶 = 𝐶𝑜𝑝𝑒𝑥+ 𝑟 ∗ 𝑅𝐴𝐵 + 𝐷 (1)

The smart grid

There is no universally accepted definition of a smart grid to be found in the existing literature (Clastres, 2011). As the name suggests, the smart grid is an intelligent grid. This implies that the smart grid is able to communicate and store information and make autonomous decisions. Silo generation, production and distribution hierarchies are replaced by fully integrated, data-driven and communicative energy systems (Farhangi, 2010). Information technology plays a major role in this, because it allows the grid to optimally use capital assets while at the same time lowering operating costs.

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losses for DSOs due to higher efficiency (National Energy Technology Laboratory, 2008). Another result of higher grid reliability is that the quality of service increases. As discussed, faults are recognized more effectively and power outages will occur less often. Higher service levels will lead to lower fines paid by DSOs and therefore operating costs will decrease.

Moreover, smart grids are able to deal with the earlier discussed distributed energy resources (DER). Active distribution network management is crucial for a cost effective integration of DG into the grid (Peças Lopes et al., 2007). Smart grids allow for this because of the flexible, interactive and predictive nature (Gharavi & Ghafurian, 2011). DER include smaller RET units, but also demand-side energy storage systems like regular batteries and electric vehicles. Energy storage is crucial, since electricity generated from RET has an intermittent character and is therefore characterized by high volatility. However, technological development is still in its infancy and efficient, cost-effective batteries are not yet available to the mass market. Therefore, it is still hard to capture the benefits of DER for DSOs and it will not be included in this analysis.

Also, smart grids provide the necessary technologies for demand response. Demand response gives consumers a chance to affect grid operations since they can shift electricity usage away from peak period and benefit through financial incentives (Tuballa & Abundo, 2016). It must be noted that active demand side management tends to alter, not reduce, electricity consumption (Palensky & Dietrich, 2011). Active demand side management might lead to a reduction in peak demand in the 27–44% range (Faruqui & Sergici, 2010). Peak load transfer is an important variable since demand for electricity is generally concentrated in the top 1% hours of the year. Therefore, ‘shaving off’ peak demand could postpone, reduce or even eliminate the need to install expensive peak capacity (Faruqui & Sergici, 2010). This implies that conventional investment in capacity enhancement can be avoided.

Table 1. Potential benefits DSOs of smart grid technologies

Technology Benefit Description

AMI Opex reduction Reduced O&M costs

AMI Opex reduction Reduced energy losses

AMI Opex reduction Avoided penalty

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4. Theory of economic regulation

Regulatory mechanisms

The standard assumption made about unregulated monopolists is that it generally sets (too) high prices and too low production. Consequently, regulation might be needed in order to incentivize DSOs to operate efficiently and prices that benefit society as a whole. Regulation has a considerable effect on investment and delays in investment potentially have enormous effects on welfare (Guthrie, 2006). However, regulation is not a perfect substitute for competition, because of the existence of information asymmetry between regulators and the regulated firm. Regulators do not have perfect information of the behavior of a regulated firm and its intentions. This implies that regulated firms generally have better information on the optimal level of distribution capacity and size of investments. Therefore, regulation is often aimed at setting boundaries and providing incentives to regulated firms (Mulder, 2016). This thesis will mainly focus on regulatory mechanisms discussed in Cambini et al. (2016). This paper identifies three categories of models: a) incentive-based models, b) cost-based models, and c) hybrid based models.

Incentive-based schemes typically let firms set their own prices. Here, a cost reduction might lead to higher profit (Vogelsang, 2002). A typical example of an incentive-based model is price cap regulation. These days, the majority of European countries use incentive-based regulatory schemes. Other examples of incentive schemes are revenue caps, yardstick competition and menus (Joskow, 2008). Cost-based models on the other hand do not provide a natural incentive for cost efficiency, because it is ensured that firms earn a predetermined rate of return (Armstrong & Sappington, 2006). Last, hybrid models combine incentive- and cost-based models. Many hybrid models treat capital expenses (CAPEX) with a cost-based approach and operating expenditures (OPEX) with an incentive-based model. Regulated revenues for a DSO are determined in the following way:

𝑅 = (1 − 𝛼)𝐶 + 𝛼𝐵 (2)

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As a matter of simplification, three regulatory models are selected. First, price cap frameworks are regarded. On the other side of the spectrum, we find rate of return regulation. This is a particular type of cost-based regulation. Somewhere in the middle of the spectrum we find yardstick regulation. Here, a distinction is made between high- and low incentive powered regulatory mechanisms. High incentive powered regulation provides natural incentives to be efficient, whereas under low incentive powered regulation firms do not have an economic incentive to increase efficiency.

With price cap regulation, an external benchmark is utilized in order to determine the maximum price a regulated firm is allowed to ask. Here, allowed revenues are adjusted according to a price index that generally includes an inflation measure and an offset for productivity gains. However, the price level is fixed for a certain period and only reviewed after his period. Price caps highly incentivize firms to increase efficiency, since the price they receive is always equal to or lower than the maximum price set. Any reduction in costs will therefore lead to higher profits. However, information asymmetry problems are present here since the regulator cannot observe the true cost function of the DSO. Since investments result in higher capital costs, total costs for the regulated firm will also increase. Given the fact that maximum revenue is fixed, firms might not be willing to invest (Mulder, 2016). The more a price cap is related to realized costs, the more certainty a DSO has that it will be reimbursed. This implies that the risk of stranded assets is higher when the external benchmark is not linked to realized costs.

Rate of return regulation gives regulated firms ex ante certainty on the rate of return they receive. In theory, rate of return regulation is regarded as an effective solution to foster investments in new infrastructure and reduce risk for DSOs, because DSOs know what return they will receive beforehand (Guthrie, 2006). However, this risk is typically shifted to the grid users, since all costs are fully passed on to these users and DSOs do not have an incentive to increase efficiency. This might lead to inefficient operating expenditures. Because the rate of return is calculated based on the regulatory asset base of a firm, rate of return regulation might lead to overinvestment. A higher regulatory asset base will namely lead to higher return for the DSO. The risk of stranded assets on the level of the DSO will be minimal, since DSOs are assured this rate of return. Allowed revenue under rate of return regulation is calculated by setting it equal to expected costs of a DSO.

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can be introduced. This means true incentive-based and cost-based models cannot solve the problem of information asymmetries itself.

Yardstick regulation might provide a solution for the information asymmetry present with price cap regulation. With yardstick regulation, the maximum level of revenues per unit is linked to the average costs of a peer group (Schleifer, 1985). A firm makes an economic profit when costs are lower than this average of the peer group and makes a loss vice versa. If DSOs will operate more efficiently due to a more productive grid (and therefore lower costs) they will have an incentive to invest. A specific type of yardstick regulation is discriminatory yardstick, where only the costs of other firms are included. Here, incentive power is at its maximum. In case the allowed revenues are based on the costs of all firms in the group, it is said to be a uniform yardstick. This is because all firms face the same revenue cap. Another feature of yardstick regulation is that all the DSO’s costs are incorporated into the tariffs. Therefore, the combined revenues and costs are equal on an industry level. On a firm level, relative productivity determines which DSO is profitable and which DSO is not. However, if DSOs do not operate in a similar economic environment as DSOs from the peer group, it might be that the average costs are too high. In this scenario, DSOs might only partially earn back their investments (Mulder, 2016).

Investments in the electricity grid are typically risky, because they might lead to stranded assets if demand does not grow as projected. Due to the costly and irrecoverable nature of these assets, DSOs fear that curbs on prices might lower return on investment. As mentioned, rate of return regulation reduces risk because DSOs know in advance what their rate of return will be. Any change in costs will be passed on to the network users. Therefore, it might be a useful regulatory mechanism to provide stability in the insecure electricity market. Nevertheless, rate of return regulation might not be optimal for smart grid investments, since investments of this type are particularly focused on efficiency enhancement. Under rate of return regulation DSOs do not benefit from a possible cost reduction as a result of smart grid investment. Price cap regulation on the other hand provides DSOs with a natural incentive to be efficient, because this will directly influence their profits. Therefore, it is expected that price cap regulation is more efficient in supporting smart grid investment.

Proposition 1. High powered incentive regulation leads to more investment than low powered

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5. Method

5.1 Investment decisions

The investment decision represents the crucial process for a firm to commit resources to future growth. However, since the firm can spend its money only once, the investment must be critically evaluated. Traditional corporate finance theory suggests that capital allocation decisions should be analyzed using discounted cash flow (DCF) models. Following this method, estimated cash flows from an investment project are discounted to their present value at a certain discount rate. If the present value of a project is positive, then the firm should invest. This discount rate reflects the market price of a project’s risk and therefore higher systematic risk will reduce the attractiveness of a project. In practice, the firm’s weighted average cost of capital (WACC) is often utilized, since it represents the firm’s required rate of return. A major criticism of the DCF approach is that it does not properly account for managerial flexibility in a project. An example of this is that managers might want to increase investment in capacity in response to higher than expected demand levels. Although the DCF method is relatively easy to apply, it is based on faulty assumptions. It assumes that investments are either reversible; or that, if an investment is irreversible, the investment decision is a now-or-never proposition. Some investment decisions indeed can be treated this way, but in most cases investments are irreversible and it might be valuable to delay investment until new information is acquired (Dixit & Pyndick, 1994).

A capital budgeting approach that accounts for the value of future flexibility is the real options approach (Trigeorgis, 1996). Real options theory is based on an important analogy within financial options. Firms with the opportunity to invest typically hold something like an American call option: They have the right, but not the obligation to invest in an asset at any time in the future. When a firm decides to invest, it basically exercises the call option. American call optionality can be present in investment decisions in multiple ways. For example, a firm might have an option to expand should conditions turn out to be favorable. As discussed, firms also have an equivalent of a call option in deciding on the optimal timing of investment.

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capacity investment is valuable for avoiding the costs of unmet demand; if future demand is unexpectedly low, the ability to mothball capacity investment avoids unnecessary expenditure.

In order to apply ROA to an investment opportunity that is the equivalent of a call option a couple of methods might be appropriate. An often employed method is developing a binomial lattice. However, the process of working through a binomial lattice can be difficult and non-intuitive, especially for more complex and high dimensional applications of real assets (Brandão, Dyer, & Hahn, 2005). Specialized Monte Carlo simulation methods have also been developed and are particularly relevant for high dimensional problems (Gamba, 2003). Real options can also be modelled using dynamic programming. It requires some mathematical sophistication and cannot be used readily for high dimensional problems. It breaks down a whole sequence of decisions into just two components: the immediate decision and a valuation function that encompasses the consequences of all subsequent decisions. Therefore, it provides an intuitive and practical approach for real options analysis. Also, a solution by contingent claims analysis can be obtained. Contingent claims analysis assumes that the investment uncertainty can be replicated by an existing portfolio (Dixit & Pyndick, 1994).

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Table 2. Glossary of Notation

Basic parameters Variables

Growth rate 𝜇 OPEX reduction due to less truck-rolls ∆𝐶𝑡𝑟

Volatility 𝜎 OPEX reduction due to electricity efficiency ∆𝐶𝑒𝑒

Discount rate price-cap regulation 𝜌𝑝𝑐 OPEX reduction due to higher quality of supply ∆𝐶𝑞𝑠

Discount rate rate-of-return regulation 𝜌𝑟𝑟 Avoided investment in capacity enhancement ∆𝐼𝑐

Share DSO in the yardstick 𝛾

Demand level 𝜃 Investment in smart technologies 𝐼𝑠𝑔

Parameters profit function

Estimated profit 𝜋̂ Estimated costs conventional grid 𝐶̂

Estimated regulated revenue 𝑅̂ Estimated costs conventional grid per unit of demand

𝑐̂

Tariff paid by consumer 𝑃̂ Estimated costs smart grid 𝐶̂𝑠𝑔

Regulated revenue under price-cap regulation

𝐵 Estimated costs smart grid per unit of demand 𝑐̂𝑠𝑔

In this thesis, a DSO that is faced with demand uncertainty is considered. Generally, a DSO has two alternatives available: investing in capacity enhancement or investing in the smart grid. Here, the option to invest in the smart grid is evaluated. Once the decision is made, it cannot be reversed. The DSO must choose at which demand level it is optimal to start investing in smart technologies. The demand level, denoted 𝜃, is assumed to follow a Geometric Brownian Motion (GBM):

𝑑𝜃

𝜃 = 𝜇𝑑𝑡 + 𝜎𝑑𝑧,

(3) where μ and σ are constants reflecting the growth and volatility rate of the demand process. As discussed, dynamic programming will be utilized in order to solve the model. A characteristic of dynamic programming is that an exogenously determined required rate of return has to be specified. This rate depends on the type of regulation considered and is classified as 𝜌. Costs (𝐶̂) are based on an estimate for a certain period and revenues (𝑅̂) are the allowed revenues determined by a regulator. Revenues are influenced by the type of regulation chosen in a country. The total expected profit (𝜋̂) of the DSO that faces the investment decision is characterized as follows:

𝜋̂ (𝜃) = 𝑅̂ − 𝐶̂ (4)

Allowed revenue per unit of demand is calculated by dividing through expected demand (𝜃𝑒). Expected demand is the demand utilized by the regulator in order to determine

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(𝑃̂) = 𝑅̂ 𝜃_𝑒

(5) Costs per unit of demand (𝑐̂) are calculated in the same way. It is the result of dividing the total cost estimate (𝐶̂) by the expected demand (𝜃𝑒). Again, this is the demand the

regulator expects in a certain period. It must be noted that regulators do not determine the total costs of the DSO.

(𝑐̂) = 𝐶̂ 𝜃𝑒

(6) When evaluating the investment decision, the DSO has an option value of waiting. This implies that it might be worthwhile to postpone investment until more, in this case about electricity demand, is known. As discussed, the techniques utilized are adopted from Dixit & Pyndick (1994). The quantity demanded is a stochastic process, which by applying dynamic programming satisfies the following differential equation:

1 2𝜎

2_𝜃2_𝑉′′_{(𝜃) + 𝜇𝜃𝑉}′_{(𝜃) − 𝜌𝑉(𝜃) + 𝜋 (𝜃) = 0,} (7)

where the primes denote the derivative with respect to 𝜃.

The value of the stage in which the DSO still has the option to invest does not include a profit flow 𝜋 (𝜃), because the DSO simply cannot get the benefits yet. Only after investing in smart technologies the DSO will be able to grasp these benefits. The value in the waiting stage therefore has the following general solution:

𝑉(𝜃) = 𝐴₁𝜃𝛽1 _{+ 𝐴} 2𝜃𝛽2 (8) where: 𝛽₁ =1 2− (𝜌) 𝜎2 + √[ (𝜌) 𝜎2 − 1 2] 2 +2𝜌 𝜎2 > 1 , (9) 𝛽₂ =1 2− 𝜌 𝜎2− √[ (𝜌) 𝜎2 − 1 2] 2 +2𝜌 𝜎2 < 0 (10)

As the level of demand reaches zero, it must be that the value of the option approaches zero too. This is because with demand reaching zero, it will become less profitable to start investing. Therefore we have boundary condition 𝑉(0) = 0. This implies that the unknown parameter 𝐴2 = 0. Therefore, the term 𝐴2𝜃𝛽2 disappears from the value function of the grid

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𝑉(𝜃) = 𝐴1𝜃𝛽1 (11)

At some moment, it is optimal for a DSO to invest in smart technologies. The value of investing in the smart grid differs per regulatory mechanism, but in general it is of the same nature as the value of not investing in the smart grid. The value of investing in the smart grid can be calculated as follows:

𝑉𝑠𝑔(𝜃) = 𝑅̂ − 𝐶̂ 𝑠𝑔 (12)

Investing in smart technologies has several benefits for DSOs. These benefits are highlighted in table 1 and are mainly aimed at cost reduction. Hereafter, these benefits will be referred to as the pay-offs of investing in the smart grid. The first pay-off of the smart grid as highlighted in table 1 is a decrease in operating and maintenance costs. Due to the use of remote diagnostics, smart grids can quickly and cost-effectively resolve maintenance issues. Effectively, this means fewer truck rolls and lower O&M expenditures for DSOs. Therefore, it is modelled as the decrease in costs for truck-rolls (∆𝐶𝑡𝑟). The cost of a truck-roll is made up

of several components and differs per utility. Generally speaking, it includes labor and truck costs. By calculating the average costs for a truck-roll and multiplying this number by the total amount of truck-rolls the total costs can be estimated.

Also, operating expenditures will be reduced due to higher efficiency of electricity usage. This means that electricity losses are reduced. The amount by which electricity losses are reduced is exogenously determined and will be multiplied by the electricity price in order to arrive at the cost savings. It is labelled as the change in operating expenditures due to higher electricity efficiency (∆𝐶𝑒𝑒). Last, DSOs will pay less penalties related to the quality of service

since the electricity grid will become more reliable. Fault detection is automated and faults in general will occur less often, which lowers these penalties. This leads to a reduction in operating expenses (∆𝐶𝑞𝑠), which is exogenously determined.

Demand response technologies can effectively postpone or even eliminate the need for conventional grid investment (∆𝐼𝑐), because electricity usage can be shifted away from

peak demand periods to periods of lower demand. This means that given the capacity at place, DSOs might be able to fulfill the electricity demand. It leads to a more flattened electricity demand curve and therefore lowers the need to invest in additional capacity. Total costs after investing in the smart grid are therefore:

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Given the above, it can be concluded that smart grid investment results in lower operating and capital expenditures. The effect of the pay-offs on the value of investing in the smart grid will vary between the different regulatory mechanisms. As mentioned, price cap regulation, rate of return regulation and yardstick regulation will be utilized as regulatory mechanisms. It is a standard result in the literature that the investment decision can be expressed as a treshold level such that whenever demand is higher than this level it is optimal to invest, and otherwise it is optimal to wait. The main principle of real options analysis is that the the value-matching and smooth-pasting conditions must be satisfied at the optimal investment trigger. The extensive solutions to the model with different regulatory mechanisms are provided in appendix A. With the treshold level denoted 𝜃∗, the value-matching and smooth-pasting conditions are respectively:

𝑉(𝜃∗_{) = 𝑉}

𝑠𝑔(𝜃∗) − 𝐼𝑠𝑔 (14)

𝑉′(𝜃∗) = 𝑉𝑠𝑔′(𝜃∗) (15)

5.2.1 Smart grid investment under price cap regulation

Price caps provide the DSO with incentives to operate efficiently, because any decrease in costs will directly influence the profit of the DSO. In this model, the DSO is not allowed price flexibility. Other authors (Dobbs, 2004) have shown that allowing the firm to have some price flexibility might improve investment efficiency. However, little empirical research has isolated the effect of price flexibility on investment timing and authors that did this found that the effect is limited. Greenstein, For example, McMaster & Spiller (1995) found that greater pricing flexibility has little impact on investment in network modernization. Therefore, it was decided that the price cap set by the regulator is the only price DSOs are allowed to charge end-users. Also, it is assumed that price cap reviews are done infrequent. With pricecap regulation (𝛼 = 1), the tariff per unit of demand becomes:

(𝑃̂) = 𝐵 𝜃_𝑒

(16)

(20)

19

part will be added fully to the DSOs profit. Consequently, the new costs per unit of demand for a DSO that invests in smart technologies under price cap regulation will be:

𝑐_𝑠𝑔 = 𝐶𝑠𝑔 𝜃_𝑒 =

𝐶 − ∆𝐶𝑡𝑟− ∆𝐶𝑒𝑒− ∆𝐶𝑞𝑠− ∆𝐼𝑐

𝜃_𝑒

(17)

This means that the expected cost reduction per unit of demand and therefore gain due to investing in the smart grid is:

∆𝐶_𝑠𝑔= ∆𝐶𝑡𝑟 + ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐 𝜃_𝑒

(18)

At the point where demand is such that it is worthwile to invest, the value in the waiting regime is equal to the value after investing in the smart grid. Explicitly, the value-matching and smooth-pasting conditions are respectively:

𝐴₁(𝜃∗)𝛽1 _{= ((}∆𝐶𝑡𝑟 + ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐 𝜃_𝑒 )) 𝜃 ∗_{− 𝐼} 𝑠𝑔 (19) 𝛽₁𝐴1(𝜃∗₎𝛽1−1 ₌ ∆𝐶𝑡𝑟 + ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐 𝜃_𝑒 (20) The value-matching and smooth-pasting conditions can be used to solve for the optimal point of investment 𝜃∗. After some rearrangement the following is found. The implications of this formula will be elaborated upon in the discussion section.

(21)

20

5.2.2 Smart grid investment under rate of return regulation

With rate of return regulation (𝛼 = 1), the tariff per unit of demand becomes: 𝑃̂ =𝐶𝑠𝑔

𝜃_𝑒

(23)

It is assumed that the review process of the regulated revenue is instantaneous, which means that the regulatory asset base is increased with the corresponding investment in smart technologies for the full amount. This also means that any operational gain is immediately transferred to the consumer and the DSO does not reap these benefits. In other words, the DSOs costs are lowered to 𝑐𝑠𝑔. Nevertheless, with the lower costs comes lower revenue, since

the revenue is determined utilizing (2). Investment expenditures can also be passed on to the consumer so these are not included in the investment decision. This means that the cost reducing effect of smart technologies does not have any value for the investment decision of the DSO. With rate of return regulation investing becomes attractive for the DSO when the regulatory asset base is expanded. This namely implies that the allowed return for the DSO increases. The allowed rate of return utilized here is the WACC (𝜌𝑟𝑟) under rate of return

regulation. The value of investing in smart technologies is therefore:

𝑉_𝑠𝑔(𝜃) = 𝜌𝑟𝑟∗ ( 𝐼𝑠𝑔− ∆𝐼𝑐) (24)

Again, the value-matching and smooth-pasting conditions need to be utilized in order to determine the optimal point of investment. These conditions are:

𝐵1𝜃∗𝛽1 = 𝜌𝑟𝑟∗ ( 𝐼𝑠𝑔− ∆𝐼𝑐) (25)

𝛽₁𝐵1(𝜃∗₎𝛽1−1 _{= 0} ₍₂₆₎

After some rearrangements, the following equation is found. The implications of this equation will be explained in the section 6.

𝜌_𝑟𝑟( 𝐼_𝑠𝑔− ∆𝐼_𝑐) ∗ 𝛽₁ = 0 (27)

5.2.3 Smart grid investment under yardstick regulation

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21

which means that it is a combination of the individual DSO’s cost function and the average costs of the other DSOs in the yardstick. Here, 𝐵 is the parameter for the average costs of the other DSOs in the yardstick. The magnitude to which the individual cost reduction or increase affects the regulated revenue depends on the share of the regulated DSO (𝛾) in the yardstick. The discount rate utilized is the one associated with price cap regulation. It was already mentioned that yardstick regulation is a form of price cap regulation, with characteristics of rate of return regulation. If the DSO is the only firm in the yardstick (𝛾 = 1), this type of regulation is similar to rate of return regulation (Mulder, 2016). The regulated revenue per unit of demand is:

𝑃̂ = 𝛾𝐶 + (1 − 𝛾)𝐵 𝜃_𝑒

(28)

Yardstick regulation is a form of incentive regulation, which means that an increase in efficiency directly influences the profit of the DSO. Due to higher efficiency, the total costs for the individual DSO that invests in smart technologies will decrease. Therefore, the decrease in costs because of investing in the smart grid will be in total added to the profit of the DSO. However, if every DSO decides to invest in smart technologies with the same benefits, the effect of the cost decrease vanishes. In this case it is assumed that the other DSOs do not invest yet. The gain per unit of demand after investing in smart technologies for the individual DSO are the same as with price cap regulation. However, the extent to which the DSO will actually benefit from it depends on its share in the yardstick. Once more, the gain due to investing in the smart grid is:

∆𝐶_𝑠𝑔= ∆𝐶𝑡𝑟+ ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐 𝜃_𝑒

(29) Since yardstick regulation is almost similar to rate of return regulation when 𝛾 = 1, the increase in expected return from an increase in regulatory asset base is also included. Therefore, the value of investing in the smart grid with yardstick regulation is:

𝑉_𝑠𝑔(𝜃) = (1 − 𝛾) ( ∆𝐶𝑡𝑟 + ∆𝐶𝑒𝑒+ ∆Cqs+ ∆𝐼𝑐

𝜃_𝑒 ) 𝜃 + 𝛾𝜌𝑝𝑐(𝐼𝑠𝑔− ∆𝐼𝑐)

(30)

(23)

22 𝐷₁𝜃∗𝛽1 _{= (1 − 𝛾) (} ∆𝐶𝑡𝑟+ ∆𝐶𝑒𝑒+∆Cqs+∆𝐼𝑐 𝜃𝑒 ) 𝜃 ∗_{+ 𝛾𝜌} 𝑝𝑐(𝐼𝑠𝑔− ∆𝐼𝑐) − (1 − 𝛾)𝐼𝑠𝑔 (31) 𝛽₁𝐷₁𝜃∗𝛽1−1 ₌ ∆𝐶𝑡𝑟+ ∆𝐶𝑒𝑒+ ∆Cqs+ ∆𝐼𝑐 𝜃_𝑒 (32)

The same as with price cap regulation, the value-matching and smooth-pasting conditions will be utilized in order to determine the optimal investment point 𝜃∗. After rearrangement, this yields the following:

( ∆𝐶𝑡𝑟+ ∆𝐶𝑒𝑒+ ∆Cqs+ ∆𝐼𝑐 𝜃_𝑒 ) 𝜃 ∗ ₌ 𝛽1 (𝛽₁− 1)(𝐼𝑠𝑔− 𝛾𝜌𝑝𝑐(𝐼𝑠𝑔− ∆𝐼𝑐) (1 − 𝛾) ) (33)

This must be solved numerically to get 𝜃∗, the investment threshold for which investment will become more beneficial than waiting. This will be done in the next section.

6. Numerical test

In this section the model is used in order to provide a numerical example in which a DSO is considering to invest in smart technologies. The investment threshold under the different regulatory mechanisms will be determined. Also, the effect of different uncertainty levels and cost savings will be assessed. The investment decision is based on a number of parameter values. These assumptions regarding these parameter values are described in the following section.

6.1 Parameter values

The value of the expected growth rate of electricity demand is 𝜇 = 0.01. This is based on the expected electricity usage growth rate estimated by the U.S. Energy Information Administration (EIA, 2017). The initial volatility of demand is 𝜎 = 0.1. As can be seen from table 2, it is assumed that the discount rate differs per regulatory mechanism. This merely is the result of higher risk under price cap regulation, which according to theory leads to a higher WACC and therefore a higher discount rate. Since price-cap regulation is seen as more risky, it was decided that the discount rate is higher than it is under rate-of-return regulation. Consequently, 𝜌𝑝𝑐= 0.05 and 𝜌𝑟𝑟= 0.025. These parameter values are based on a report

(24)

23

Since the estimated cost reductions discussed in this model are very specific for an individual firm, a simplifying assumption is made. Here, the reduction in operating expenditures is determined as a result of automated metering infrastructure (AMI). All operating expenditures are namely related to the utilization of AMI and therefore it provides a sensible estimate of the decrease in expenditures. Mott MacDonald (2007) estimated that the operating expenditures will be reduced by approximately 7%. Due to demand response technologies, investment in capacity enhancement can also be avoided. Because data on the avoided investment in capacity enhancement was not present it is assumed that this is ten million euro. Similarly, no real data was available for the total amount of the investment in the smart grid, but it is set at twelve million in order to account for insecurity in the development of the smart grid costs. In the real world, investment in capacity enhancement is often capital intensive and therefore one might expect that it is even higher than the investment in the smart grid. Expected demand was set at ten million and the initial cost level that is used in the calculations amount to fifty million. Even though these values seem rather arbitrary, they do not influence the results since the parameter values utilized are equal for all three regulatory mechanisms. With these parameter values investment thresholds of 14.11 million and 13.64 million are found for price cap regulation and yardstick regulation respectively. Hence, under the given parameter values the investment threshold with yardstick regulation is lower.

Table 3. Parameter values

Basic parameters Variables

𝝁 0.01 𝐶𝑜𝑝𝑒𝑥 50 𝝈 0.1 ∆𝐶𝑜𝑝𝑒𝑥 3,5 𝝆𝒑𝒄 0.05 𝐶̂𝑠𝑔 46,5 𝝆𝒓𝒓 0.025 ∆𝐼𝑐 10 𝜸 0.5 𝐼𝑠𝑔 12 𝜃𝒆 10 6.2 Effect of uncertainty

(25)

24

Figure 1: Investment thresholds as function of the volatility in demand with 𝛾 = 0.5 and 𝐼𝑠𝑔= 12

Figure 1 indeed shows that the investment thresholds go up as the volatility increases. Also, it can be seen that in the entire interval the investment threshold of price cap regulation is under the investment threshold of yardstick regulation. So when comparing both forms of high powered incentive regulation, it is shown that for the given parameter values yardstick regulation leads to a slightly lower investment threshold for DSOs. This is mainly because the DSO benefits from a cost reduction, but also partially earns back the investment costs. There is a clear trade-off between the amount to which investment costs are earned back and the way in which the DSO benefits from the cost reduction. When the share of the firm in the yardstick is larger (𝛾 = 0.8), the difference between price cap regulation and yardstick regulation becomes more prominent as can be seen in figure 2.

(26)

25

So as the share of the firm in the yardstick increases, yardstick regulation will more likely promote investment in smart technologies compared to price cap regulation for the given investment expenditure. Price cap regulation will be favorable when the investment costs are lower, as presented in figure 3. This is because earning back investment costs will be easier when they are lower, given the same benefits of investing in the smart grid. This seems rather straightforward, but it has some implications that will be discussed in section 7.

6.3 Effect of higher cost reduction

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26

Figure 4: Investment thresholds as a function of the percentage of OPEX reduction with 𝛾 = 0.5 and 𝐼𝑠𝑔= 12

Figure 4 shows that given the parameters discussed in section 6.1, there is almost no difference between price cap regulation and yardstick regulation. As the reduction in operating expenditures grows, basically the power grid becomes more efficient. So with relatively high efficiency yardstick is still more likely to give investment incentives, but the difference is minimal. Similar to the inferences made in section 6.2, the difference between price cap regulation and yardstick regulation becomes larger as the share of the firm in the yardstick increases. However, it must be noted that this difference decreases as the reduction in operating expenditures becomes larger. This is mainly due to the higher incentive power of price cap regulation. Still the amount to which investment costs can be passed on to network users determines the gap between price cap regulation and yardstick regulation.

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27

However, when the investment expenditures become lower price cap regulation is more likely to promote smart grid investment. Again, with price cap regulation DSOs have a higher incentive for cost reducing investments, because they will fully benefit from these reductions. This is not the case for yardstick regulation. As investment costs decrease, the benefit of passing on investment expenditures becomes less prevalent. Therefore, price cap regulation becomes more favorable. This situation is visualized in figure 6.

7. Discussion

In the section 5, a model is provided that assesses the benefits and costs of investing in smart technologies. It has been made clear that the different regulatory mechanisms give the firm different incentives to invest. As elaborately discussed, with price cap regulation the DSO is highly incentivized to increase efficiency since all costs savings are retained within the firm. Therefore, any increase in efficiency will lead to higher profit. This is a bit different from rate of return regulation, where according to the model the only factor that influences the investment decision is whether the regulatory asset base is expanded. This seems intuitive, because a higher regulatory asset base implies that the DSOs allowed rate of return increases. Yardstick regulation has characteristics of both price cap regulation and rate of return regulation. If the share of the firm in the yardstick is high, yardstick regulation is rather similar to rate of return regulation. Therefore, the decision to invest depends on the expansion of the regulatory asset base. In case the share of the DSO in the yardstick is low, the DSO is more incentivized to invest in efficiency enhancement. However, it must be noted that if all DSOs in the yardstick invest the gain from investing in smart technologies will be lower. This is because

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28

the sum of the total costs will then be decreased too and the allowed tariff a DSO can ask depends on the total costs in an industry.

The model developed in section 5 is checked in section 6 given certain parameter values. Unfortunately rate of return regulation could not be visualized in the same graph as the other two regulatory mechanisms. However, as the share of the firm in the yardstick increases, yardstick regulation will become more similar to rate of return regulation. Therefore, it can be treated as yardstick regulation with a high share in the yardstick and inferences can be drawn based upon this. As shown in section 6 the investment threshold indeed increases as uncertainty increases, for both price cap regulation and yardstick regulation. Therefore, the increased uncertainty will increase the value of waiting and postpone the point of investment. However, for the given parameter values in section 6.1 the difference is rather limited as can be seen in figure 1. This difference becomes bigger when the share of the firm in the yardstick is bigger, since then more of the investment expenditure can be recouped. From (27) it is found that uncertainty does play a role in rate of return regulation, but it only increases the value of investing since 𝛽1 can never be smaller than zero.

As uncertainty increases, 𝛽₁ will go down to one. However, this only holds when the investment in the smart grid is higher than the avoided investment in capacity enhancement.

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29

in capacity enhancement, yardstick regulation should be preferred. One might even say that rate of return regulation should be preferred, since the higher the share of the firm in the yardstick, the lower the investment threshold. However, this argument has to be enriched with more evidence, which is not provided in this thesis.

Figures 4-6 represent the effect of a reduction in operating expenditures on the investment threshold. Alternatively, one might say that it represents the effect of higher efficiency on the investment threshold. Figure 4 shows that, using the parameters described in section 6.1, the difference between price cap regulation and yardstick regulation is limited. As the share of the firm in the yardstick increases, yardstick regulation becomes more attractive when investment costs remain higher than the avoided investment. This is similar to what was found for the relationship between the uncertainty level and the investment threshold. However, as investment costs decrease a strong effect is demonstrated. Here, the combination between higher cost reduction and a lower investment threshold leads to price cap regulation being favourable. The threshold level will even drop beneath the expected level of demand utilized by the regulator to determine the tariff for both price cap regulation and yardstick regulation.

In conclusion, the share of the firm in the yardstick and the costs of investing in the smart grid influence the choice of the regulatory mechanisms. Here, price cap regulation will be favourable when investment costs are lower than the avoided investment in capacity enhancement. However, when this is not the case yardstick regulation is the preferred regulatory mechanism. When investment costs are higher than the avoided investment, an increase in the share of the firm in the yardstick leads to a lower investment threshold when the investment in the smart grid increases.

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8. Conclusion & Limitations

The concept of the smart grid is evolving. A lot of attention has been paid to it lately and it is expected that this attention will only be growing. A much heard voice is that the problem for shifting to a more sustainable energy supply is not a matter of production, but of transmission in distribution. The current centralized electricity grid needs to undergo tremendous change and this transition requires substantial investment. This thesis has attempted to provide a decision model in which investment thresholds under different regulatory mechanisms are determined. This is highly relevant, since DSOs still operate in a regulated environment. First, the benefits of investing in the smart grid were discussed, after which these benefits were modelled using a real options approach. This approach was chosen as the valuation method since investments in the smart grid are characterized by high uncertainty. In this thesis, demand uncertainty is the source of uncertainty considered.

It has been shown that this uncertainty indeed postpones the decision to invest. When volatility increases, so does the demand threshold level. This result supports the literature and proves that the model correctly represents the investment opportunity. It was found that for the given parameters in section 6.1, yardstick regulation leads to a slightly lower investment threshold than price cap regulation. Unfortunately, rate of return regulation could not be visualized, since it was concluded that the investment threshold does not matter here. Under rate of return regulation only an increase in the regulatory asset base will namely incentivize a DSO to invest. It was shown that proposition one is true, depending on the way in which the investment costs of the smart grid are evolving. In case the investment costs are lower than the avoided investment in capacity enhancement, price cap regulation will lead to a lower investment threshold. However, when the investment expenditure is higher than the avoided investment in capacity enhancement, yardstick regulation will give a higher incentive to invest. This is also the case when the decrease in operating expenditures, resulting from higher efficiency, becomes larger. One might expect that price cap regulation is the preferred regulatory mechanism because all efficiency gains are captured here, but it was shown that this is not true.

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grid. Obviously, the study can be expanded by including more tariff mechanisms. Moreover, the model developed in this thesis is subject to one source of uncertainty. Nevertheless, investing in the smart grid can be affected by multiple sources of uncertainty. Here, one could think about uncertainty in the development of the smart grid investment expenditure or uncertainty in the way operating expenditures decrease. However, including more sources of uncertainty results in more complex and high dimensional applications and requires specialized Monte Carlo simulation models. Hence, a future study that includes multiple sources of uncertainty might provide a broader understanding of the investment opportunity. Furthermore, the investment problem handled in this thesis focusses on investment in the smart grid. However, it might be the case that demand evolves as such that investment in capacity enhancement cannot be avoided. This might be a very interesting topic for future research, since it enriches the investment model presented in this thesis.

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Appendix

Appendix A: Full solution to investment models

Solution with price cap regulation

The investment threshold will be solved using the value-matching and smooth-pasting conditions. The smooth-pasting condition is the derivative of the value-matching condition with respect to 𝜃∗. The value-matching and smooth-pasting are the following respectively:

𝐴₁(𝜃∗₎𝛽1 _{= ((}∆𝐶𝑡𝑟 + ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐 𝜃_𝑒 )) 𝜃 ∗_{− 𝐼} 𝑠𝑔 𝛽₁𝐴₁(𝜃∗)𝛽1−1 ₌∆𝐶𝑡𝑟+ ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐 𝜃_𝑒

The equations stated above have to be solved for the demand level 𝜃∗. In order to do this, the value-matching condition is multiplied by 𝛽1 and the smooth-pasting condition is multiplied by

𝜃∗_{. This yields the following equations:}

𝛽₁𝐴₁(𝜃∗)𝛽1 _{= 𝛽} 1(( ∆𝐶𝑡𝑟 + ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐 𝜃_𝑒 )) 𝜃 ∗_{− 𝛽} 1𝐼𝑠𝑔 𝛽1𝐴1(𝜃∗)𝛽1 = ( ∆𝐶_𝑡𝑟+ ∆𝐶_𝑒𝑒+ ∆𝐶_𝑞𝑠+ ∆𝐼_𝑐 𝜃𝑒 ) 𝜃∗

Now, the left terms of both equations are the same, so the right-terms can be equalized. This gives: 𝛽₁((∆𝐶𝑡𝑟+ ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐 𝜃_𝑒 )) 𝜃 ∗_{− 𝛽} 1𝐼𝑠𝑔= ( ∆𝐶_𝑡𝑟+ ∆𝐶_𝑒𝑒+ ∆𝐶_𝑞𝑠+ ∆𝐼_𝑐 𝜃_𝑒 ) 𝜃 ∗

After rearrangement, this gives:

(𝛽1− 1) ((

∆𝐶_𝑡𝑟 + ∆𝐶_𝑒𝑒+ ∆𝐶_𝑞𝑠+ ∆𝐼_𝑐 𝜃𝑒

)) 𝜃∗ = 𝛽1𝐼𝑠𝑔

This leads to the following solution:

(∆𝐶𝑡𝑟 + ∆𝐶𝑒𝑒+ ∆𝐶𝑞𝑠+ ∆𝐼𝑐

𝜃_𝑒 ) 𝜃

∗₌ 𝛽1

THE IMPACT OF TARIFF REGULATION ON SMART GRID INVESTMENT