A Lattice Method for Jump- Applied to Transmission Expansion

Investments under Demand and Distributed Generation Uncertainties

Fikri Kucuksayacigil a, K. Jo Min b a University of Arkansas, Fayetteville, AR, [email protected] b Industrial and Manufacturing Systems Engineering Department, Iowa State University, Ames,

IA

Corresponding author: Fikri Kucuksayacigil, 4179 Bell Engineering Center, University of

Arkansas, Fayetteville, AR, [email protected], 515-203-9808

Abstract

Strategic decision making for rare events is considered to be an arduous course of actions as their impacts cannot accurately be modeled. Transmission of electrical power is a major challenge in this day and age due to the prevalent of rare events. Decision makers of generation units, distribution utilities, and transmission networks do not often cooperate, which creates severe uncertainties for all players. Growth of demand for electricity and installation or removal of distributed generation (DG), considered to be a rare event, are among uncertainties encountered by transmission network owners. Expansion decisions for transmission lines should be strategically executed because installations of DGs may create a stranded cost for transmission owners. In this study, we propose a real options framework that quantifies the values of transmission investments under demand and DG uncertainties to guide decision makers of transmission companies regarding how to adapt their expansions decisions. We model demand uncertainty as a geometric Brownian motion (GBM) process and DG uncertainty as a Poisson arrival process. We devise a computationally-efficient lattice diagram to discretize both processes in a single grid, which can also be employed to model other types of rare events in various sectors.

The proposed framework is demonstrated on a realistic transmission network. Numerical study shows that our proposed is computationally superior over existing diagrams. As for managerial insights, it shows that depending on the locations of DG installations, DG penetration may not reduce the value of a transmission network. If the center at which a DG is installed has a large-capacity local generator, the installation may not undervalue the transmission network.

Keywords: Rare events, distributed generations, jump-diffusion process, lattice framework, real options, transmission expansion investments

1 Introduction

After deregulation in the electricity market of the United States, decision makers of transmission companies / transmission owners confront critical uncertainties when they make investments. The reason is that transmission owners may not have prior information of decisions made by generation and distribution companies or communities. The trend in electricity demand is one of the severe uncertainties because it may exhibit large fluctuations [1]. Installation of distributed generations (DGs) creates another uncertainty because they have potential to defer large-scale transmission expansion investments.

When transmission owners are not informed of DG installations, transmission investments tend to be redundant. Experts in the electricity market have initiated discussions toward evaluating the impact of DGs on costs and benefits of transmission investments. They point out that transmission investments could be better-planned if the rate of future adoption of DGs could be correctly estimated [2]. Interested readers can also see [3] for a discussion concerning transmission investment redundancy associated with a badly-designed distribution network. DGs have been installed to meet local demand for electricity during the previous decade in various sizes ranging

2 from only a couple of megawatts to tens of megawatts. Reference [4] shows a summary data listing various DG technologies preferred by utilities and societies as well as capacities installed in each year from 2006 to 2015.

Transmission investments usually require huge capital costs, causing strategic decision making to be indispensable, and presence of uncertainties turns this problem to even more intractable form.

A modeling framework to quantify the values of transmission investments in the presence of such uncertainties is a vital requirement. By values of transmission investment, we mean their monetary value, which is estimated with their future profits subject to an uncertain decision environment.

This description is actually the definition of real options value of investments. In this study, we propose and demonstrate a real options approach to show how transmission investments can be assessed under continuous (infinitesimal changes in infinitesimal time interval) and discrete uncertainties (random shocks or rare events), namely uncertainties of growth in electricity demand and DG installations and removals. While a wide array of studies has been conducted for evaluating transmission investments under electricity demand uncertainty, the evaluation of DG penetration is a fairly new area of research. Most studies model DG installation uncertainty with discrete penetration scenarios, but our model stands unique because it uses a Poisson arrival process. It has a significant advantage over defining discrete scenarios because it is sufficient to determine only one parameter, arrival rate. As it will be seen in mathematical formulations, the only parameter as to uncertainty of DG penetration is arrival rate. As soon as we determine this parameter, we capture the corresponding uncertainty. Poisson distribution is an approximation of a counting process which would normally be modeled with a binomial distribution. First-order approximation enables us to suffice with arrival rate. In discrete penetration scenarios, one needs to know what scenarios are and what they have. It is challenging to estimate both

3 simultaneously with an acceptable accuracy. If we had perfect information about the distribution of uncertainties, we would be able to generate all relevant distributions.

Investment evaluation problems modeled with real options methodology can be solved through three alternative approaches. While analytical techniques are worthwhile because managerial insights do not depend on numerical values of parameters, dealing with an analytically tractable model often requires advancing many unrealistic and restrictive assumptions. Monte Carlo simulation, proposed as one alternative for American options, provides researchers with modeling flexibilities such as ability of handling with jump and diffusion processes without enforcing a sequence between processes. On the other hand, it has a significant drawback from a computational perspective. For example, [5] ran through 50,000 paths to obtain an average value of American options for each of the stochastic processes underlying stock prices they were interested in modeling. Finally, with respect to lattice methods, it is often claimed that they require much less computation time compared to Monte Carlo simulation while returning more accurate results [6].

With lattice methods, however, one may not be sure about stability of results.

In this study, we take advantage of lattice frameworks to model demand growth. As will be seen in Sect. 3, estimating the revenue of a transmission network requires solving an optimization problem. This problem is impractical to embed in an analytical model, which is the fundamental reason why we do not use an analytical approach We also want to avoid simulation due to its expensive computation time. There are numerous studies in the energy literature describing discretizing continuous evolutions with lattice or tree diagrams to solve problems without sacrificing their core properties [7 - 9]. Merit of our lattice model is that quantification of transmission network values is quickly accomplished. Although our framework finds the basis in a dynamic programming tree, often challenged because of its curse of dimensionality, we mitigate

4 this problem using approximation techniques, proposed in this study. As will be seen in next sections, our lattice model discretizes both diffusion and jump processes. Whereas diffusion process in our case is random smooth growth of demand for electricity, is random installation of DGs in a power network.

This study is structured as follows: In Sect. 2, we present the literature on work in close proximity to our paper. We then identify a discrete counterpart of the underlying uncertainty path in Sect. 3 and introduce a method to reduce its computational complexity. We also show how we quantify transmission network values. This is followed by a numerical example in Sect. 4 in which we demonstrate our framework on a realistic transmission network and show computational superiority of our proposed lattice model over existing ones. In Sect. 5, we discuss some assumptions made throughout the paper and present generalizations of our framework. Sect.

6concludes the paper by summarizing key points and important managerial insights.

2 Literature review

A rich body of literature on analyzing transmission investments has been presented. Since we approach this problem by considering DG uncertainties, we restrict our attention to studies that incorporate DG penetrations with transmission investments. The following is a quick overview of those studies.

Hejeejo and Qiu [10] examine transmission investments and power networks with intermittent DG resources. Investments are put in place to meet peak electricity demand while maintaining network reliability. The random output of DG is accepted as an uncertainty creating an impact on transmission investments. This study treats DG as an already-established resource in the network.

Our study differentiates itself in the sense that it models the uncertainty of new installations of

DGs in the network. Rouhani et al. [11] investigate integrated generation and transmission

5 expansion problems using a proposed multi-step approach. In the first step, types of generating units are determined so as to minimize the cost of production. The second step involves allocation of new generation units and transmission lines to the network along with deciding the locations of

DGs. Our paper is different in the sense that a transmission owner is solely responsible for transmission investments. Zhao et al. [12] study the impacts of DG penetration on the deferral of transmission expansion investments. They model DG penetration as certain percentages in overall power supply that need to be met by DGs. In a different study, Gomes and Saraiva [13] assume that a DG would account for 0%, 10%, 15%, and 20% of annual peak demand in each bus and observe their influences on transmission expansion investments. Similarly, Rathore and Roy [14] consider that DG capacities can be 15% or 20% of demand for electricity at each bus in a given power network. Moradi et al. [15] likewise estimate that DG in the form of wind farms may penetrate to a power network by accounting for either 10% or 20% of total energy supply. Our study is different than these work to model DG penetration with Poisson arrival process and to adjust its penetration level through arrival rates.One study similar in nature to ours was conducted by Luo et al. [16]. Investment in the network is assumed to be performed by a transmission network operator who also controls generation expansions. Penetration of DGs representing the percentages of their shares in electricity generation are modeled through a few scenarios. Their random outputs are also modeled with Weibull and normal distributions for wind and solar generations, respectively. The main theme of this study is to examine the deferral effect of DGs on transmission investments (see also [17] for deferral effect of energy storage on transmission upgrades). A similar study is published by Sarid and Tzur [18]. They design a power network with generation and transmission expansions as well as DG presences modeled through varying penetration levels.

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In contrast to these studies, we model DG penetrations with Poisson arrival processes rather than discrete scenarios.

There is a group of studies that regards DG as a tool for a transmission owner to use in postponing large-scale network upgrades. Vasquez and Olsina [19] employ a real options framework to quantify the values gained by DGs in deferring options. Distribution network owners also have accepted DGs as flexibility tools [20 - 22]. This literature differs from our paper in that we do not assume DGs are policy tools for transmission network owners.

In summary, our paper becomes unique with the consolidation of following aspects: (i) We assume that DGs are not installed when the value of transmission network is estimated. Instead, we state that DGs can be installed in future with a probability. We also assume that if DGs are already- installed in the network, they can be removed in future with a probability. (ii) Some of the papers reviewed above approach transmission expansion problem with the assumption that DGs are investment options for decision makers. Our paper considers that DGs installation or removal is completely an uncertainty for transmission network owner, which translates that network owners do not have any control on DG investments. (iii) Our paper stands unique with the modeling of

DG uncertainty. While abundant number of papers model DG installation uncertainty with discrete penetration scenarios, we model it with Poisson arrival process. (iv) Although it is not listed in the literature review, we reiterate that our paper develops computationally less expensive lattice model to estimate the values of transmission investments, which should be emphasized within the unique aspects of our paper.

In the next section, we first present discrete counterparts of underlying uncertainties, followed by explaining a naïve idea related to reducing computational complexity of discrete models. At the

7 end of the section, we demonstrate a framework for quantifying the values of transmission networks.

3 Mathematical model

In energy literature, growth of electricity demand is usually assumed to follow a geometric

Brownian motion (GBM) process [23, 24]. Marathe and Ryan [25] also validate this assumption empirically..

Since installation or removal of a DG at a consumption center alters demand for electricity formerly met by transmission lines, a smooth path of demand (GBM) is punctuated by abnormal jumps to higher or lower levels. Occurrence of such jumps and their magnitudes, which are arrival of DGs and their capacities in our context, are random, thus uncertainty of DG installations can be modeled with compound Poisson arrival process [26]. Interested readers can see Sect. 5 for discussions about these assumptions and their possible generalizations. Incorporation of GBM with a compound Poisson arrival is called a jump-diffusion process in the literature. To reiterate, jump is random installations of DGs with random capacities, and diffusion is random growth of electricity demand modeled with GBM in our context.

Note that this description of jump-diffusion process is for a single consumption center. In reality, a transmission network is comprised of numerous consumption centers. Thus, for such networks, we need to use a multi-dimensional jump-diffusion process which isa combination of individual jump-diffusion processes.

There is a rich body of literature in financial pricing that develops discrete (lattice) counterparts of jump-diffusion processes [26 - 28]. We build our framework on the model proposed by Hilliard and Schwartz [29], a single-dimensional jump-diffusion process that we

8 extend to a multi-dimensional jump-diffusion process. We afterward reduce the computational complexity of the extended model.

3.1 Discretization of jump-diffusion process

3.1.1 Lattice model of jump-diffusion process for a single consumption center

Hilliard and Schwartz [29] discretize a jump-diffusion process by matching its local moments with discrete branches. Discretized form of the process is given by

±휎√∆푡 푏ℎ 퐷푡+1 = 퐷푡e e (1)

where 퐷푡 is demand for electricity at time point 푡 (MW), 휎 is of demand evolution

(%/unit time), and ∆푡 is the length of a period in the lattice (period 푡 spans from time point 푡 to time point 푡 + 1). ±휎√∆푡 represents up or down movements of the diffusion process. Jump magnitude, denoted by 휅 referring to percentage change in 퐷푡, is assumed be to a random variable following a log-normal distribution [30] with parameters (훾, 훿2). To discretize it, 푏 takes on values of {0, ±1, ±2, … , ±푚}, meaning that it is discretized on 2푚 + 1 points. The difference between successive jump nodes in the vertical order is denoted by ℎ = √훾2 + 훿2. Note that the number of

DG events (installations or removals) are controlled by a with arrival rate 휆 (the number of events per unit time). The risk-neutral probability of an up movement in the diffusion process (+휎√∆푡) is given by

휎2 1 1 푟 − 휆휅̂ − 푝 = + ( 2 ) √∆푡 (2) 2 2 휎

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1 훾+ 훿2 where 푟 is risk-free interest rate (%/unit time) and 휅̂ = E[휅] − 1 = e 2 − 1. Probabilities of jump branches, denoted by 푞(푏), are found by matching 2푚 moments of the jump process. To summarize,

푞(−푚) 1 1 ⋯ 1 ⋯ 1 1 −1 푞(−(푚 − 1)) 1 1 1 1 1 (−푚) (−(푚 − 1)) ⋯ 0 ⋯ (푚 − 1) 푚 휇 ⁄ℎ ⋮ 1 2 2 2 2 2 푞(0) = (−푚) (−(푚 − 1)) ⋯ 0 ⋯ (푚 − 1) 푚 휇2⁄ℎ (3) ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 푞(푚 − 1) 2푚 [휇 ⁄ℎ2푚] [(−푚)2푚 (−(푚 − 1)) ⋯ 0 ⋯ (푚 − 1)2푚 푚2푚] 2푚 [ 푞(푚) ]

calculates 푞(푏) where 휇1 is the first moment of jump process, 휇2 is the second moment, and so on.

When jump magnitude is not random, Equation (1) is adjusted with 푏 = 1 and replacing ℎ with 훾,

휎√∆푡 훾 i.e., 퐷푡+1 = 퐷푡e e . For more details of discretization procedure, see [29].

3.1.2 Lattice model of jump-diffusion process for multiple consumption centers

푖 Let 푖 denote a consumption center in a transmission network and let 퐷푡 denote demand of this

푖 center at time 푡. 퐷푡 has the same parameters 휎푖, 휅푖, 휅̂푖, and 휆푖 as introduced in previous section.

Probabilities of branches, 푝푖 and 푞푖(∙), have the same expressions given in Equations (2) and (3) with parameters ℎ푖 and 휇푖. Let |푁퐷| denote the set of consumption centers in the network. In this case, the diffusion part of the demand lattice, which is the first increments in Equation (1), is turned into a 2|푁퐷|-branch lattice by considering all combinations of up and down movements of diffusion processes. Wang and Min [31] give joint risk-neutral probability of an arbitrary branch 푙 as

|푁퐷| |푁퐷| |푁퐷| ′ 1 푝푙 = ∏ 푝푖 + ∑ ∑ 푦푖푗휌푖푗 (4) 2|푁퐷| 푖=1 푖=1 푗=푖+1

10

푖 푗 |푁퐷| ′ where 휌푖푗 is the correlation coefficient between 퐷푡 and 퐷푡 and 푙 = 1,2, … , 2 . 푝푖 and 푦푖푗 are given by

′ 푝푖, if process 푖 moves upward in 푙 푝푖 = { (5) 1 − 푝푖, if process 푖 moves downward in 푙

1, if processes 푖 and 푗 move in the same direction in 푙 푦 = { (6) 푖푗 −1, if processes 푖 and 푗 move in the opposite direction in 푙

Each individual diffusion has its own jump. Just as generally assumed in the literature, we assume jump process is independent of its diffusion and other jumps. We propose to incorporate jumps into multi-dimensional diffusion and adjust probabilities. With the addition of jump branches, the overall probability of the increment from time point 푡 to 푡 + 1 is calculated as the joint probability

(denoted by 퓅) of diffusion and jump branches (Fig. 1 and Fig. 2). For example, in Fig. 1 the

푖 푗 푖 휎푖√Δ푡 ℎ푖 푗 −휎푗√Δ푡 ℎ푗 probability of (퐷푡 , 퐷푡 ) to reach (퐷푡 e e , 퐷푡 e e ) is 퓅 = (푝푖푝푗 + 휌⁄4)푞푖(1)푞푗(1).

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Fig. 1 Demand evolution lattice for two consumption centers with random jump magnitude, 푚 =

1.We only show the jump branches emanating from the second diffusion branch for the sake of expositional convenience. We show diffusion and jump processes separately although it is not a requirement. We draw them separately for expositional convenience as well as due to the fact that we do not know which process moves first. Probabilities of jump branches are given at the right.

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Fig. 2 Demand evolution lattice for two consumption centers with fixed jump magnitude.

Probabilities of jump branches are given at the right.

3.1.3 Computationally relaxed lattice model

Lattice models shown in Fig. 1 and Fig. 2 are not useful due to curse of dimensionality. It is obvious that a modeling horizon involving a few periods leads to an immense number of states in the lattice.

To mitigate it, we propose a relaxation procedure. Instead of allowing jump events to happen at every period,as in the original model, Fig. 3, we allow them to be realized at every 푣 periods,proposed model, Fig. 4. We claim that, given a small value of 휆, the probability distributions of the terminal nodes in the original and proposed lattices approximate one another.

As it will be obvious in the rest of the section, 푣 is a parameter to strike a balance between computational needs and the reflection of the real world. Small 푣 reflects the real world better with a computational expense. Yet our claim is that with small 휆, the real world will still be reflected in a sufficient level even if 푣 is large.

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Fig. 3 Original lattice with a single diffusion and jump processes. Jump process is discretized in three nodes. Only a small subset of branches is demonstrated due to expositional convenience.

This is the initial part of the whole lattice, meaning that branches continue emanating from terminal nodes such as D and E as well.

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Fig. 4 Proposed lattice with a single diffusion and jump processes. Jump events happen at every three periods.

Notice that the lattices illustrated in Fig. 3 and Fig. 4 are initial parts of entire lattices, implying that more branches emanate from nodes labeled with D to I. These continuum lattices are structurally the same as the initial parts, which we call recurring structure. In fact, we use this characteristic to demonstrate the approximation mathematically. Due to the recurring structure, it is sufficient to show the approximation only at terminal nodes of initial parts of entire lattices.

To mathematically justify the approximation, let’s simplify the lattice models by considering single diffusion and jump processes with fixed jump magnitude. In the original lattice, demand values at the terminal nodes (at time point 푣 + 1) have the general expression

푢 푑 휎√∆푡 −휎√∆푡 훾 푥 퐷0 (e ) (e ) (e ) . It means there exist 푢 up movements and 푑 down movements of the diffusion process as well as jump events happen 푥 times in 푣 periods. The probability at this

15

푣 푣 푢( )푑( ⁄ )푥( ⁄ )푣−푥 ⁄ node is ℙ표푟푖𝑔푖푛푎푙 = (푢)(푥)푝 1 − 푝 휆 푇 1 − 휆 푇 where 1 푇 is equal to 훥푡. We can make the following observations:

푥  If 푥 ≥ 2 and 휆 is small enough, (휆⁄푇) → 0. Hence, ℙ표푟푖𝑔푖푛푎푙 approaches to 0.

푣 푢( )푑( ⁄ )( ⁄ )푣−1  If 푥 = 1, ℙ표푟푖𝑔푖푛푎푙 turns into (푢)푣푝 1 − 푝 휆 푇 1 − 휆 푇 . Note that when 휆 is

( ⁄ )푣−1 푣 푢( small enough, 1 − 휆 푇 → 1. Thus, ℙ표푟푖𝑔푖푛푎푙 can be written as (푢)푝 1 −

푝)푑(휆⁄푇)푣.

푣 푢( )푑( ⁄ )푣  If 푥 = 0, ℙ표푟푖𝑔푖푛푎푙 is summarized as (푢)푝 1 − 푝 1 − 휆 푇 . Notice again that when

( ⁄ )푣 푣 푢( )푑 휆 is small enough, 1 − 휆 푇 → 1, so ℙ표푟푖𝑔푖푛푎푙 approaches (푢)푝 1 − 푝 .

Note that proposed lattice model up to time point 푣 is the same as the binomial lattice of [32], in which probabilities of up and down movements of diffusion process are not functions of 휆. This does not create a problem since we can neglect 휆 in Equation (2) because it is sufficiently small.

In this case, we can still use Equation (2) as our probability expression, reconciling with the probability equation given in [32] with 휆 neglected.

In the proposed model, demand values at the terminal nodes have general expressions of

푢 푑 푢 푑 휎√∆푡 −휎√∆푡 훾 휎√∆푡 −휎√∆푡 퐷0 (e ) (e ) e and 퐷0 (e ) (e ) depending on whether a jump event occurs.

The respective probabilities of these values are given as follows:

푗푢푚푝 푣 푢( )푑( ⁄ ) ⁄  ℙ푝푟표푝표푠푒푑 = (푢)푝 1 − 푝 휆 푇 푣, where 휆 푇 is the probability of jump event to occur

in a period and (휆⁄푇)푣 is the probability of jump event to occur in 푣 periods.

푛표 푗푢푚푝 푣 푢( )푑( ( ⁄ ) ) ( ( ⁄ ) )  ℙ푝푟표푝표푠푒푑 = (푢)푝 1 − 푝 1 − 휆 푇 푣 , in which 1 − 휆 푇 푣 → 1 when 휆 is

푛표 푗푢푚푝 푣 푢( )푑 sufficiently small, so ℙ푝푟표푝표푠푒푑 approaches (푢)푝 1 − 푝 .

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푗푢푚푝 Notice that ℙ표푟푖𝑔푖푛푎푙 for 푥 = 1 approaches ℙ푝푟표푝표푠푒푑. Similarly, ℙ표푟푖𝑔푖푛푎푙 for 푥 = 0 approaches

푛표 푗푢푚푝 ℙ푝푟표푝표푠푒푑. With this approximation, it can be inferred that one can use the proposed lattice model rather than the original version because resulting values of transmission networks will also approach one another. A reasonable level for 푣 can be selected by the modeler. Small values of 푣 give rise to more accurate results with the risk of having unreasonably long computation times.

It is reasonable to employ the proposed lattice model in a DG context because installations or removals of DGs are not events that happen frequently, implying that their corresponding arrival rates are small.

3.2 Quantification of values of transmission investments

It is intriguing that there is no consensus concerning the remuneration system related to transmission investments in the electricity market [33]. . States or electric power markets in the

U.S. have different policies depending on market structures in their corresponding service areas.

Some areas adopt a pure regulatory approach where investment cost plus a rate of return is assured for a transmission owner. Others insure a part of investment cost with a fixed revenue and complete the remainder by allowing transmission owners to collect variable revenue depending on locational marginal price (LMP) differences. To the best of our knowledge, no state in the U.S. adopts a pure merchant business model. PJM (Pennsylvania-New Jersey-Maryland Interconnection) is a good example for electric power markets providing two components of revenue. It provides market participants with an opportunity to collect revenue based on LMP differences. In this market, power-generating companies also pay for transmission investments that can be regarded as fixed part of revenue (for a more enlightening discussion, see [34]). We observe that some other electricity markets have designed a system in which distribution utilities and power generators pay

17 for transmission investments (see [35, 36] for an example of transmission access charge). This type of revenue for transmission companies is often called supplementary revenue because it is considered to be financially impractical to allow transmission owners to collect revenue solely based on LMP differences. Since transmission investments sometimes relieve network congestion and decrease LMP differences, the decline in revenue should be compensated with a supplementary revenue [37].

In this study, for a hypothetical transmission company we model the revenue with LMP differences in the network. We do not analyze a pure regulatory business model because transmission companies do not possess strategic flexibilities. In the case of an investment, we allow a supplementary revenue for transmission owners [24]. Notice that, although the company is hypothetical, it is realistic in the sense that our framework is consistent with systems adopted by various electric power markets such as PJM.

Our framework is separately conducted for each investment alternative that is defined as addition of a power line between two arbitrary centers. We first consider the case of no investment in the network, which we call base case. Each state in the demand lattice is used to compute the net present value (NPV) of the network as a state variable. A new lattice demonstrating the evolution of network value is created for the base case by state-by-state mapping between demand and NPV lattices. We then proceed to evaluate each investment alternative. Since an investment can be postponed by the decision maker, we separately take into account different time points of investment (choices). Choice 푡 corresponds to the investment made at time point 푡 for the selected investment alternative. For each choice, a different lattice showing the evolution of network value is created.

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LMP is the local price of electricity ($/MWh) computed by solving an optimal power flow (OPF) problem. As we consider day-ahead segments of the electricity market, we calculate LMP hourly

[38, 39]. The LMP-based revenue of the network, denoted by 푅 ($/hour) is calculated by

푅 = ∑ 휋푖퐷푖 − ∑ 휋푗퐺푗 (7) 푖∈푁퐷 푗∈푁퐺

where 휋푖 denotes LMP at center 푖, 퐺푗 denotes dispatched amount of generation center 푗 at optimality of the OPF problem (MW), and 푁퐷 and 푁퐺 denote the set of consumption and generation centers (푁퐷 ∪ 푁퐺 = 푁), respectively. The OPF problem is stated as

min ∑ 푤푖퐺푖 (8) 푖∈푁퐺

퐺푖 − 퐷푖 = ∑ ℬ푖푗(휃푖 − 휃푗), ∀푖 ∈ 푁 (9) 푗∈푁,푗≠푖

−풷푖푗, if 푖 ≠ 푗 푁 ℬ푖푗 = { (10) ∑ 풷푖푗 , otherwise 푗∈푁,푖≠푗

−퐿푖푗 ≤ ℬ푖푗(휃푖 − 휃푗) ≤ 퐿푖푗, ∀(푖, 푗) ∈ 푀 (11)

0 ≤ 퐺푖 ≤ 퐺̅푖, ∀푖 ∈ 푁 (12)

where 푀 is the set of power lines, 푤푖 is generation cost of generation center 푖 ($/MWh), ℬ푖푗 is a matrix consisting of actual susceptance values (in Siemens), 풷푖푗 is susceptance of the power line between centers 푖 and 푗, 휃푖 is voltage angle at center 푖 (in Radians), 퐿푖푗 is the capacity of the power line between centers 푖 and 푗 (MW), and 퐺̅푖 is the capacity of generation center 푖 (MW). Notice that

19 we use linearized OPF problem formulation as opposed to typical nonlinear version. See Sect. 5 for a discussion concerning this issue.

LMP at center 푖 is obtained as follows. The OPF problem is solved with given demand values and an objective function value is recorded. The problem is then resolved with demand value increased by 1 MW at center 푖. The new objective function value minus its previously recorded value gives the LMP at center 푖. Alternative approaches to compute LMP are existing in the literature.

Interested readers can review Sect. 5 to see our discussion in this issue.

3.2.1 Quantification of transmission network value for base case

In the lattices, we denote a node with subscripts 푡 (time points) and 푘 (states), with the value of 푘 starting from 1 at the uppermost node and incrementing by 1 through the bottommost node for each 푡.

Valuation starts with terminal nodes of the lattice. We assume that the network is removed at the end of the modeling horizon (푡 = 풯), incurring a decommissioning cost denoted by 𝒞 ($). Hence, the value of the network at time point 풯 is the negative of the decommissioning cost. At time point

풯 − 1, discounted total profit is calculated for a ∆푡 length of time by making the assumptions that profit is realized at the end of each time period and demand does not change during ∆푡. In other words, discounted total profit gained during Δ푡 duration is

8760(푅(풯−1,푘) − ℂ)Δ푡 푃 = (13) (풯−1,푘) 1 + 푟

where 푅(풯−1,푘) is network revenue calculated with Equation (7), and ℂ is operation and maintenance cost ($/hour). NPV of the network at time point 풯 − 1 ($) is finally defined as

20

𝒞 푉 = 푃 − (14) (풯−1,푘) (풯−1,푘) 1 + 푟 by taking into account the discounted decommissioning cost. For the other intermediate nodes (푡 <

풯 − 1), discounted risk-neutral expected value of the successor nodes is added after calculating the profit with Equation (13). In other words,

푉(푡,푘) = 푃(푡,푘) + ∑ 퓅푙푉(푡+1,푘) ⁄1 + 푟 (15) 푙∈푆(푡,푘) ′ (푘∈푆(푡,푘) )

′ where 푆(푡,푘) denotes the set of branches emanating from (푡, 푘) and 푆(푡,푘) denotes the set of successor states of (푡, 푘). 푉(1,1), obtained through recursive computation in Equation (15), is accepted as the network value for the base case.

3.2.2 Quantification of transmission network value with an investment

In the case of an investment, there are 풯 − 1 choices for timing and different NPV lattices are created for each choice using Equations (13), (14), and (15). When an investment is carried out at the beginning of period 푡, a supplementary revenue 퐹 ($) and an investment cost 퐼 ($) are incorporated. If the investment is made at time point 풯 − 1, then

𝒞 푉 = 퐹 − 퐼 + 푃 − (16) (풯−1,푘) (풯−1,푘) 1 + 푟

If the investment is made at an arbitrary time point 푡 < 풯 − 1, then

21

푉(푡,푘) = 퐹 − 퐼 + 푃(푡,푘) + ∑ 퓅푙푉(푡+1,푘) ⁄1 + 푟 (17) 푙∈푆(푡,푘) ′ (푘∈푆(푡,푘) )

For Choice 푡, we estimate the value of investment by subtracting 푉(1,1) calculated for the base case from 푉(1,1) calculated for the network with the investment made at time point 푡. If this difference is negative, the investment value is regarded as 0.

4 Numerical example

In this section, the developed framework is demonstrated through a simple numerical example inspired by a hypothetical transmission network given in the PJM training materials [40]. The PJM example consists of five centers, three generation centers and three consumption centers (one common center), modified for our own purposes. Interested readers can see [38, 41] for the same example with modifications. In the following section, we elaborate steps we take to make the hypothetical example as realistic as possible.

4.1 Justification of numerical example

The transmission network is a simplified representation of a realistic system with only a few centers. It is not a complete abstraction from reality because the parameter values are chosen as proxies for real values. While we obtain solutions and derive managerial insights based only on this reasonably realistic network, they are also applicable to large-scale systems. Real power networks with abundant number of generation centers do not require much computation time to run our model as they do not worsen the curse of dimensionality. Yet large number of consumption centers in networks and more granularity in binomial lattices will suffer from the curse of dimensionality and they will demand more time to run our model. Large number of consumption

22 centers in power networks will necessitate that DG installation or removal will be modeled for these centers which increase the number of jump branches in lattices. Note that our model finds its superiority in mitigating this complexity.

Fig. 5 A transmission network in North Virginia (PJM region)

There are three centers in the network (Fig. 5), each connected to another with a single power line.

While this is a representation of a transmission network in the Dominion Energy service area (PJM market) located in Virginia, not all data used is perfectly matched with that system. The centers, also known as bus names in PJM interactive transmission map and located in Morrisville,

Loudoun, and Ox counties, serve populations living in the Remington, Loudoun, and Lake Ridge regions with low-voltage transmission lines, usually less than 230-kV. An interactive transmission

23 map shows that Loudoun is connected to Morrisville and Ox with 230-kV transmission lines and

Morrisville is connected to Ox with a 500-kV transmission line. We assume that 230-kV transmission lines have a total capacity of 450 MW while the 500-kV line has a capacity of 800

MW. These values are not abstractions from reality [42]. Since our exhaustive search has not provided us with data regarding the capacities of newly added investment lines, we will simply assume that added lines in all three circuits have 100 MW capacity. The length of transmission lines are 26 miles, 39 miles, and 41 miles for the circuits connecting Loudoun - Ox, Loudoun -

Morrisville, and Ox - Morrisville, respectively. In this numerical example, we assume that susceptance of both existing and newly added power lines are identical (풷12 = 풷13 = 풷12 = 1) although our framework is generalizable to accept varying susceptance values.

There are two generation centers with capacities of 1300 MW each located in the Morrisville and

Loudoun regions. These capacity values are realistic because generators within short distances of

Morrisville and Loudoun have varying capacities, e.g., 780 MW of Panda Stonewall Energy Center and 1700 MW of nuclear power stations [43]. We assume that generation centers in Morrisville and Loudoun have respective generation costs of $35/MWh and $20/MWh. These values are reasonable because PJM historical day-ahead and real-time LMP data indicates that the most likely LMP values range between $20/MWh and $30/MWh. Furthermore, operational data given for the PJM website shows that the Dominion service area has LMP values such as $18/MWh and

$22/MWh.

Demand for electricity in Morrisville and Ox are 800 MW and 500 MW, respectively. These values make sense because a similarly-populated region has a demand of 429 MW [44]. Reference [25] analyzed historical trends to estimate volatility of demand growth as 0.0096.Using this value in our numerical example is afflicted with a problem, i.e., since our modeling horizon is fairly short,

24 such small volatility would not create significantly-varying levels of demand over periods. Thus, we adjust this value to a more plausible level and assume volatility of demand to be 0.07 per year.

We also simplify the problem by considering zero correlation with demand growth. While this does not change problem structure, it eases the analysis. We also assume that DGs can be installed at consumption centers. One challenging DG parameter to estimate is arrival rate, and we adopt a rough approach to make such an estimation. Dominion Energy [45] indicates that, since in 2015 there were 30 solar panels in Virginia that began to be installed, the overall DG arrival rate 7.5 per year. To the best of our knowledge, other forms of DGs such as wind power are not available options in Virginia. The PJM interactive transmission map points out that there are approximately

30 buses connected with 500-kV transmission lines, leading us to the conclusion that that a region has an arrival rate of 0.24 DG per year. Dominion Energy [45] also indicates that capacities of

DGs vary widely, ranging from 0.05 MW to 200 MW, so our capacity assumption 150 MW is realistic. For the sake of simplification, we assume that DGs have fixed capacities. Note also that we only consider DG installations, not removals, to simply the problem.

With respect to the cost parameters of our example, [46] indicates that $959,700 and $1,919,450 were the respective costs of investing in 1 mile of 230-kV and 500-kV transmission lines, and transmission lines were decommissioned at a net cost of $155,000 per 1.4 mile [47]. We learned from [48] that operation and maintenance costs for transmission lines were $4,771 in 2010 dollars per circuit mile, convertible to today’s dollars as $5,487 using the inflation rate. Notice that we increase decommissioning and operation and maintenance costs in the presence of an investment.

American Municipal Power, Inc. [49] shows that transmission rates, which are possibly regarded as supplemental revenue in the case of an investment in our framework, are enormously high throughout the PJM region, in the order of $50,000 MW-year for Dominion Electric. Considering

25 the fact that it sold more than 60 million MWh in 2007[50]; transmission access charges turned out to be billions of dollars. Similarly, transmission access charges in California tend to be immense [36]. In our framework, we treat supplemental revenue as multiples of corresponding investment costs, e.g., 1.1 times investment cost.

Finally, the risk-free discount rate is assumed to be 0.021 per year because [51] lists 10-year nominal interest rate for riskless bonds as 0.021. We chose a modeling horizon of 2 years with a period length of 0.5 year. We wanted to keep modeling horizon relatively shorter because, as will be seen in CPU times, running original lattice model for even 2 years takes enormous amount of time. Increasing modeling horizon, to 5 years for example, without changing the granularity level would cause the models to demand exponentially more computation times, as the number of nodes in demand lattices would skyrocket exponentially. In Sect. 4.2, we use lattice model proposed in this study with jumps occurring only once over modeling horizon. In Sect. 4.4, we compare performances of original and proposed lattice models. The algorithms were implemented in Matlab

R2017b and run on a remote server with 252.2 GiB and 2.6 GHz Intel Xeon CPU E5-2660 v3 processor.

4.2 Network values without DG and with DG presence

In this section, we present network values for three cases: (i) There is no uncertainty in DG installations or removals, (ii) a DG has a chance to be installed at consumption center 1, and (iii) a DG installation may be realized at consumption center 3. These three circumstances constitute mutually exclusive events in such a network with only two consumption centers, which is why we pay attention to these cases. By analyzing the distinctions in results, we will have an opportunity to observe the effect of separate DG installations at different locations. Note that for each of these cases, there are four choices for timing of investments, at the beginnings of each of the four periods.

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Tables 1, 2, and 3 list respective 푉(1,1) values for the base case and for investments with different investment times for each of these cases.

Table 1 푉(1,1) without DG

푉(1,1) of Investments ($) Choices Between Centers Between Centers Between Centers Base Case 1 and 2 1 and 3 2 and 3 No Investment 413,292,976 - - - 1 - 205,478,673 346,446,655 424,670,318 2 - 248,619,640 361,105,094 423,089,548 3 - 291,312,332 375,611,215 421,525,202 4 - 333,561,401 389,966,780 419,977,112

Table 2 푉(1,1) with DG presence at consumption center 1

푉(1,1) of Investments ($) Choices Between Centers Between Centers Between Centers Base Case 1 and 2 1 and 3 2 and 3 No Investment 413,292,974 - - - 1 - 205,478,678 346,446,683 424,656,806 2 - 248,619,645 361,105,121 423,076,035 3 - 291,312,338 375,611,242 421,511,690 4 - 333,561,406 389,966,808 419,963,599

Table 3 푉(1,1) with DG presence at consumption center 3

푉(1,1) of Investments ($) Choices Between Centers Between Centers Between Centers Base Case 1 and 2 1 and 3 2 and 3 No Investment 394,174,285 - - - 1 - 205,478,675 335,825,157 417,574,445 2 - 248,619,642 350,483,596 415,993,674 3 - 291,312,334 364,989,717 414,429,329 4 - 333,561,403 379,345,282 412,881,239

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We also examine the value of transmission network with changing volatilities, arrival rates, and capacities of DGs. Fig. 6(a) represents the value of a transmission network for the three cases defined above over a small range of the volatility parameter. Fig. 6(b) and Fig. 6(c) show the evolution of transmission network values for two cases with a reasonable range of arrival rates and capacities of DGs. We will discuss Tables, 1, 2, and 3 as well as Fig. 6 in the next section.

(a)

(b)

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(c)

Fig. 6 Sensitivity of transmission network value with respect to volatility (a), arrival rates (b) and capacities of DGs (c)

4.3 Results and discussions

푁표 퐷퐺 퐷퐺 푎푡 푐푒푛푡푒푟 1 푉(1,1) values for base cases in Tables 1, 2, and 3, designated as 푉(1,1) , 푉(1,1) , and

퐷퐺 푎푡 푐푒푛푡푒푟 3 퐷퐺 푎푡 푐푒푛푡푒푟 1 푉(1,1) , convey significant managerial insight. It can be observed that 푉(1,1) is not

푁표 퐷퐺 less than 푉(1,1) . It would be expected that installation of a DG would most likely undervalue transmission lines because the community with a DG is only partially in need of the lines. Our results contradict this expectation and emphasize that center 1 is redundant with respect to determining the dispatch amounts of generation centers. Whenever demand marginally increases at this center, it is met by its own generation plant. Although a DG is installed at center 1, the network still produces a high level of revenue because of the high demand at center 3. Note that this observation is not limited merely to base case comparison. In fact, Table 1 and Table 2 are almost identical, implying that a DG at center 1 would be ineffective in undervaluing the transmission network regardless of base case or presence of investments. Fig. 6(b) and Fig. 6(c)

29 also demonstrate that massive changes in arrival rate and capacity of DG at center 1 do not impact transmission network value. Notice also that network values without a DG and with a DG at center

1 are almost equal according to Fig. 6(a).

퐷퐺 푎푡 푐푒푛푡푒푟 3 푁표 퐷퐺 With respect to a DG at center 3, 푉(1,1) is significantly less than 푉(1,1) because a huge demand reduction impacts the dispatches of both generation centers at distance. This observation is also applicable to investment cases, except between centers 1 and 2. This investment sufficiently relieves the congestion so that the same set of LMPs are still generated. From Fig. 6(b) and Fig.

6(c), we can infer that transmission network value is highly sensitive to a change in arrival rate and capacity of a DG at center 3. Fig. 6(a) conveys the idea that network value with a DG at center

3 is always less than that without a DG.

Fig. 6(a) also supports the idea that, as volatility of demand increases, transmission network values reduce. It is known that risk can have either positive or negative effects in investments, and in our case, risk has a downside effect on transmission network values.

Investments between centers 1 and 2 and center 1 and 3 are delayed to the end of the modeling horizon. Since new lines in these circuits decrease LMP-based revenues, a decision maker may attempt to gain higher revenue by not adding a power line earlier. In contrast, investments between centers 2 and 3 are conducted at the beginning of the first year because LMP-based revenues turn out to be higher throughout the first year. When the investments are conducted at this particular period, the investments create more congestion in the network and make more profit.

As noted, investments between centers 1 and 2 and centers 1 and 3 diminish the value of transmission network. For each of Tables 1, 2, and 3, we observe that values of these investments in all investment times are always less than their corresponding base network values. It implies

30 that these investments should not be considered at all as there does not exist any profit to do so. If they are the unique alternatives, then they should be executed at the end of modeling horizon. On the other hand, values of investment between centers 2 and 3 are always larger than their corresponding base value. Hence, decision maker should choose this pathway to add a new transmission line. Since more profit should be desirable, addition of a new line should be executed at the beginning of modeling horizon.

Question may arise as to the validation of these results. We argue that validation of these results is a demanding process and could take a long time. We would like to emphasize that our primary focus in this paper is the modeling of a challenging problem and provide suitable solution procedures. For validation process, given that a real-world transmission network is chosen, one needs to aggregate historical data for DG capacities and evolution of their installation and removal patterns. Historical data of demand for electricity should also be aggregated. Generation costs, their historical bidding behaviors, and generation capacities should also be collected. Then, investment alternatives for transmission network expansion should be determined, ideally with the help of expert knowledge. In the final stage, our proposed computational framework are ready to be executed. We list validation of results in our future research agenda.

4.4 Computational performances of original and proposed lattice models

In this section, we will briefly address the computational advantages of the proposed lattice model.

For this analysis, we will take into account simultaneous installations of DGs at consumption centers 1 and 3. We will also change the arrival rates of DG installations from 0.24 per year to

0.05 per year in order to prevent numerical inconsistencies. Simultaneous installations of DGs with relatively larger values of arrival rate may give rise to infeasible values of branch probabilities in the lattices. That is the reason why we should be judicious in selecting numerical values of arrival

31 rates. In reality, since there exist abundant number of centers in a power network, data perturbation will have a limited influence on bringing about infeasible values. . Table 4 reflects the results of this analysis.

Table 4 푉(1,1) values with original and proposed lattice models

Original lattice model Proposed lattice model Modeling Difference 훥푡 푉 for Computation 푉 for Computation horizon (1,1) (1,1) Between (years) base case Time base case Time (years) 푉( ) (%) ($) (seconds) ($) (seconds) 1,1 2.5 0.5 481,621,162 84,784 488,322,341 318 1.372 2 0.5 405,039,701 5,328 409,391,256 80 1.063 1.5 0.5 325,527,607 323 327,933,927 20 0.734 1 0.5 245,157,349 22 246,216,435 6 0.430 0.5 0.5 161,483,296 1.97 161,483,296 1.86 0

Table 4 clearly shows that approximately the same transmission network values are obtained with two lattice models, but with an immense difference in computation times. As we increase the modeling horizon, we expect to see computation time of original lattice model to skyrocket whereas proposed model performs with a slight increase in computation time. We can sketch a simple demonstration for time complexity of two models: Recall that |푁퐷| is the number of consumption centers in transmission network, 2푚 + 1 is the number of branches to discretize a jump process, and 푇 is the number of periods in the modeling horizon.

푇 Original lattice model will have (2|푁퐷|(2푚 + 1)|푁퐷|) nodes at the end of modeling horizon in demand lattice. 2|푁퐷| is the number of diffusion branches and (2푚 + 1)|푁퐷| is the number of jump branches in a period. Overall in 푇 periods, the number of branches is equal to

푇 (2|푁퐷|(2푚 + 1)|푁퐷|) .

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In proposed lattice model, recall that jump branches appear once in 푣 periods. Thus, there is no jump branch in 푣 − 1 periods. The number of diffusion branches in 푣 − 1 periods is much less than 2|푁퐷|(푣−1) due to recombining property of diffusion branches. Interested readers can see Fig.

4 for recombining property. One would expect that there will four nodes at the end of period 2, but it turns out that there are three due to recombining branches, also known as path-independency.

This is the reason why the number of diffusion branches is 푣 − 1 periods is much less than

2|푁퐷|(푣−1). In period 푣, there will more 2|푁퐷|(2푚 + 1)|푁퐷| nodes. Thus, at the end of 푣 period, there will be ≪ 2|푁퐷|푣(2푚 + 1)|푁퐷| nodes. Finally, at the end of 푇 periods, there will be ≪

1 푇 | | |푁 | (2 푁퐷 (2푚 + 1) 퐷 푣) nodes. It proves that the number of nodes at the end of 푇 periods in proposed model is significantly less than that of original model.

In summary, there are two gains through our proposed model. Even without recombining property

(no ≪ in above expressions), computational complexity of the proposed lattice model is less than that of original model with exponent 1⁄푣 for (2푚 + 1)|푁퐷| because jump events appear once in every 푣 periods. With the recombining property, the second gain, computational complexity of the proposed model is even much less than that of original model. As the number of periods in the modeling horizon increases, we observe expanding effect of recombining property. Hence, as the number of periods grows, we see difference between computation times of two models gets larger.

5 Discussions about assumptions, modeling framework, and numerical example

In this section, we address some issues requiring more thoughtful analysis, explanations, and generalizations.

Arrival of DGs: We make some implicit assumptions when we model DG installations or removals through a Poisson arrivals process. In general, the process implies that the probability of one arrival

33 between 푡 and 푡 + 훥푡 is independent of 푡. While we make this assumption in the DG context, it may pose a challenge because it is empirically evident that more DG installations have been observed in recent times. More appropriate models such as a non-homogeneous Poisson arrival process might be used to track the change in arrival rates. Another property of a Poisson arrival process is that it is a viable modeling option when arrivals originate from a large population of independent communities [52]. In the DG context, we see no intuitive reason for dependent communities to install DGs. In other words, since it is not likely to find that the probability of DG installation in a particular region affects the probability of DG installation in another region, even without empirical evidence we think that DG arrivals or removals can be soundly based on a

Poisson arrival process. Another implicit but less critical assumption is independence of a Poisson arrival process from a diffusion process. If this was not a valid assumption, the probability of increase in electricity demand would impact the probability of DG installations. We do not think that this is not the case. To us, it seems not reasonable to envision a customer who decides on installing a DG as his/her electricity demand/consumption is high. We expect DG installations are likely incentivized by electricity price and customer bills, not by demand. If people pay skyrocketing price for their electricity, they more likely install DGs even if their consumption is small. That is why it is not wrong to assume independence of a Poisson arrival process and a diffusion process. This assumption is also a common practice in jump-diffusion process (see seminal work [26 - 30]). Assuming dependency between jump and diffusion process would be analytically intractable. In fact, we had left this as our future research. As noted, jump processes are independent of each other as well. We state that it is not reasonable to assume one region will install a DG solely because the other region installed it, which refers to dependency of jump processes. DG installations in different regions will be incentivized by their own consumer

34 behavior and customer bills. That is why it is not erroneous to assume independency of jump processes. Similarly, this is also a common assumption in jump-diffusion literature, see [26].

Dependency between jump processes is listed in our future research agenda.

Underlying uncertainty: Our framework can be generalized in multiple ways. Although it describes diffusion evolution with GBM because historical demand data fits GBM process, this is not a strict or essential requirement. Binomial, trinomial, or multinomial lattices are employed to model other types of continuous stochastic processes such as mean-reverting and two-correlated

Ornstein-Uhlenbeck (see [53] for a comprehensive discussion). Moreover, our framework does not necessarily need to adopt demand for electricity as an underlying uncertain parameter. Fuel cost or policy changes could be comfortably embraced in a jump-diffusion type of evolution. Fuel cost (fuel prices, spot price of oil, etc.) has been empirically proven to follow a GBM process [54,

55], so policy changes can be modeled as Poisson process because they have potential to extensively impact fuel prices.

Uncertain output of DGs: Another generalization can be made in the following sense: Note that while our framework works for both random and fixed capacities of DGs, we do not deal with the output uncertainty of DGs. Typical DGs such as small-scale wind turbines and solar photovoltaics are known to be intermittent. In this framework, we neglect this aspect of DGs because its involvement with our framework makes the problem much more challenging. However, ignoring this aspect is not an absolute abstraction from real-life applications. Recent technological advancements enable energy to be stored in a wide array of available forms. We assume that when demand for electricity is low, energy can be stored and subsequently used to meet increasing demand. We are lead to think that demand for electricity by communities with DGs decreases by their capacities. Even though uncertain output of DGs can be simulated without technical

35 difficulty, we refrain from this real-world feature in our framework for the sake of exposition clarity.

Quantification of network values: Note that underlying theme of our framework is to quantify the values of transmission expansion investments by modeling their revenues. Our study differentiates from other published studies that model the same problem from the perspective of cost minimization [56, 57]. Such studies usually embrace investments and operation and maintenance costs and involve them into an objective function to be minimized. Optimization models are typically solved with linear, mixed-integer linear, and stochastic programming techniques. In this paper, we consider revenue of a transmission investment as objective, and we quantify investment values with a stochastic dynamic programming approach, using Equation (7) as revenue for our framework. Alternative formulations in the electricity markets have been advanced [58] and they can be incorporated into our framework with little effort.

OPF problem formulation: Equations (8) - (12) is the OPF problem formulation, a linearization of direct current flow approximation. It is sufficient and legitimate to use this formulation to model power flow in high-voltage transmission lines [59]. Long lines along with high-voltage capacities enable us to disregard alternative current parameters such as reactive power, trigonometrical relations, and transmission losses [60]. Interested readers can find more details regarding approximations and linearization of OPF problem in [61, 62]. Note that we use minimization of total generating cost of electricity in the objective function. Independent system operators conventionally solve the OPF problem with maximization of total social welfare being the sum of consumer and producer surplus. In this study, we assume consumers are price-insensitive, and in such a case, the maximization of total social welfare is equivalent to minimization of total generating cost of electricity. Notice also that the OPF problem formulation has a linear cost

36 function for generators. While the degree of cost function is not a central theme in our paper, it can be modified into any form of cost function without difficulty. We omit reliability and security issues of transmission networks in this paper because they are not in the core of our analysis.

LMP calculation: The way we compute LMP is a layman’s definition, as used practically in electricity markets. LMP can be also calculated through shadow prices / values of Lagrange multipliers of Equation (9) [63]. These approaches may not give rise to the same set of LMPs because shadow prices, by definition, are calculated from the infinitesimal change on demand values [64], and it is obvious that an increase of 1 MW is not an infinitesimal change. To reconcile the situation, we recalculate LMPs using the layman’s definition and increasing demand values by a small amount such as 0.1 MW. We find that the two sets of LMPs, calculated by the layman’s definition and by Lagrange multipliers, are equal in this case.

Equality of susceptance: Notice that even though power-carrying capacities of transmission lines may be different, we assume equality of susceptance, a legitimate assumption for the following reasons: the thermal limit of a power line is in reality limited to so-called surge impedance loading, the proportion of end bus voltage to characteristic impedance of a power line [65]. Since characteristic impedances of sufficiently long power lines are approximately equal, susceptance of power lines can be freely taken as equal as long as they are long enough, even though they may have different capacities. There is also a rich body of literature that makes this simplifying assumption [66]. In this respect, we suggest a seminal working paper by J. Bushnell and S. Stoft entitled as ‘Transmission and generation investment in a competitive electric power industry’ written in 1995 at University of California Energy Institute.

Small-scale numerical example: A question may arise as to why a small-scale example is preferred to demonstrate the framework. To tell the truth, this is a typical approach in the literature

37 where the problem of interest is elaborately analyzed, aiming to focus on modeling aspects rather than on computationally-expensive analysis. Conveying the idea with small-scale examples is significant present in a vast body of energy literature [67]. Interested readers can also see a working paper by R. Kapuscinski, S. Suresh and O. Wu entitled as ‘Operations and investment of energy storage in the presence of transmission losses’ written in 2015 at Indiana University Bloomington.

A three-center power network is a particularly popular structure used to demonstrate the economics of transmission investments, LMP calculations, and dynamics of financial transmission rights [66,

68, 69]. Lumbreras and Ramos [70] also state that dynamic aspects of some problems makes them challenging, and it is often preferred to show key aspects of modeling frameworks on small-scale numerical examples.

6 Conclusion

To conclude the paper, we have proposed a real options framework to quantify values of transmission investments under demand and DG uncertainties. We modeled underlying uncertainties with GBM and compound Poisson processes, and made use of a lattice approach for discretion. We proposed an idea of reducing computational complexity using the lattice model, and key components of the framework have been demonstrated through a numerical example. The results indicate that decision makers should not make a priori judgements that network value decreases when a DG is installed. If installation locations are redundant with respect to determining dispatch amounts of generation centers, DGs may have little effect on the value of transmission lines. Furthermore, investments may either relieve or increase network congestion, which determines whether investments should be done as soon as or as late as possible. Future studies could involve two paths. First, correlation between GBM and compound Poisson processes could be taken into account. When demand for electricity increases, there may be higher likelihood of

38

DG installations. Second, correlation between multiple compound Poisson processes could be considered. A community may prefer a DG if a neighbor community installs it because they may have identical intentions.

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