A Lattice Method for Jump-Diffusion Process 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 lattice model 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 probabilities 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 probability 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, jump process 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