A statistical learning approach to determine optimal sizing & investment timing of commercial-scale distributed energy resources

https://doi.org/10.1016/j.scs.2020.102596Get rights and content

Highlights

  • Optimal size and time for investment in combination of micro-grid assets is presented.

  • The problem of optimal sizing is solved using dynamic programming.

  • Investment time problem is formulated using real option approach.

  • RPART decision trees are developed to estimate the optimization models output.

  • Sensitivity analysis is done to evaluate the performance of original and robust model.

Abstract

The problem of optimal sizing and investment timing for a portfolio of gas-fired and photovoltaic (PV) generation is presented and solved using an optimization- classification technique. It is assumed that the demand, the prices of natural gas, and PV technology are stochastic processes. Our methodology takes advantage of decomposing the problem into two sub-problems. First, the optimal sizing problem is solved using dynamic programming and most likely solutions are identified as clusters. Then, the Investment time problem is formulated as a Real Option for each cluster to determine optimal timing. Although the two-step optimization approach can successfully close the loop between operational dynamics and investment decisions, we are also interested in discovering patterns in a multidimensional space of input parameters that make a certain combination of assets optimal among dozens of discrete choices. On that note, by applying Recursive Partitioning algorithm, decision trees are developed to estimate the structure of solutions rendered from optimization models by a rule-based system. Despite the high level of accuracy, the initial model is biased in favor of highly frequent clusters, and discards the optimal clusters resulted from extreme market behaviors. Value at Risk (VaR) is employed as a risk measure to demonstrate the enhancement risk performance. Finally, in order to investigate the robustness of the results, we conduct extensive sensitivity analysis over different parameter settings. The proposed model can be thought of as a statistically optimal summarizer of optimization models that enables decision makers to have an insight into optimal investment strategies according to characteristics of the building and the long-term energy market outlook.

Introduction

The problem of interest is to determine optimal sizing and investment timing of a combination of assets during a planning horizon with the goal of maximizing savings from the distributed generation over the lifetime of the project, which is considerably higher than the planning period.

In this paper our contribution in the first place is to demonstrate a way of solving real option problems regarding commercial scale behind-the-meter generation investment with discrete choices (referred to as clusters in this paper). Second, we show that the two complex stochastic optimization problems can be consolidated into ruling systems, in form of decision tree structures, that determine the optimal size and time of investment based on a handful of input parameters. Lastly, we devise a weighting algorithm to address the issue of bias towards highly frequent clusters which hedge the decision trees against the abnormalities in the market of influencing factors. Value at Risk (VaR) is employed as a risk measure to demonstrate the misclassification risk reduction.

The block diagram below (Fig. 1) exhibits the workflow of this paper. We take a practical approach that allows us to separate the timing and sizing decisions. In phase I (chapter 2.2), we compute the optimal sizing over a set of stochastic inputs. Upon doing so, we found that the majority of optimal sizing solutions are not out of a small set of “clusters” (combination of assets) across all the stochastic paths. This is an interesting finding, since it allows us to apply real option analysis on optimal investment timing to each of these clusters separately in phase II (chapter 2.3). Next, we emulate the functionality of optimization models by generating decision trees which take the parameters of each Monte Carlo path as an input and estimate the optimal cluster in their leaves (chapter 3). And ultimately (chapter 3.3), we explain how the RPART tree can be tweaked to increase robustness. Finally (Chapter 4), we examine our models against multiple variations in assumptions. While the decomposition of large and complex stochastic optimization problems into simpler problems is not new, our optimization-classification approach warrants some serious attention as it gives useful insight into how system input and output behave. In particular, it seems that at least in some specific circumstances, it is possible to switch from a functional view of input data into a more behavioral view. This is more in line with what a domain expert would do intuitively. It is also worth noting that while regular portfolio analysis procedures commonly base their strategies upon weak assumptions about the interaction of investment parameters, thanks to the comprehensive simulations in phase I, the proposed decision support system can precisely evaluate the financial takeaways resulting from each bundle installments of assets.

Due to high dimensionality of energy investment problem, a vast number of organizations and experts have strived to devise integrated decision support systems (DSS) that help decision makers to find the ideal energy assets based on a selection of variables. (Masini & Menichetti, 2012) proposes and tests a conceptual model that examines the structural and behavioral factors affecting the investors’ decisions. The proposed model demonstrates how the investors’ a-priori beliefs, their preferences over policy instruments, and their attitude toward technological risk affects the likelihood of investing in renewable energy projects. (Wüstenhagen & Menichetti, 2012) introduces processes underlying strategic choices for renewable energy investment and how they are influenced by energy policies, and discuss the Segmentation of financial investors along the innovation chain. (Kumar et al., 2017) along with (Pohekar & Ramachandran, 2004), (Wang, Duanmu, Lahdelma, & Li, 2017), (Baudry, Macharis, & Vallée, 2017), (Haralambopoulos & Polatidis, 2003), (Cai, Huang, Lin, Nie, & Tan, 2009) review and apply various Multi Criteria Decision Making (MCDM) techniques to assess progress made by considering renewable energy applications and future prospects in this area. To overcome the inconsistency inherited in qualitative measurements, (Franco, Bojesen, Hougaard, & Nielsen, 2015) and (Kahraman, 2010) apply Fuzzy set theory to address the information imprecision problem in estimation of the criteria weights of AHP methodology. (Kyriakarakos, Patlitzianas, Damasiotis, & Papastefanakis, 2014) presents the design and implementation of a fuzzy cognitive maps (FCM) decision support toolkit (DST) for local renewable energy sources planning.

Narrowing down to the economic aspects of Distributed Energy Resources (DER) investment, the main problem related to the integration of microgrid systems into the electricity grid comes from its intermittency and unpredictable nature of resources, and it is prudent to incorporate the uncertainty in future conditions. To that end, a myriad of articles have been scripted to serve this issue in variety of scales and from different standpoints. (Li, Coit, & Felder, 2016) addresses the importance of climate change impact on Generation Expansion Planning problem by presenting expected total cost and maximum regret minimization from risk-neutral and risk-averse perspectives.(Lin, Yang, & Xu, 2020) Optimization models considers different evaluate indexes, the part load performance of equipment and different operation circumstances in different demand in CHP systems revenue generations. (Kuznetsova, Ruiz, Li, & Zio, 2015) presents an analysis of a microgrid energy management framework based on Robust Optimization (RO) and optimization based on expected values. Uncertainties in wind power generation and energy consumption are described in the form of Prediction Intervals. This work also shows how the probability of occurrence of some specific uncertain events, e.g., failures of electrical lines and electricity demand and price peaks, highly conditions the reliability and performance indicators of the microgrid. To reduce the uncertain influence of solar photovoltaic power in the scale of power plants operation, (Ju et al., 2016) utilizes robust optimization to build a stochastic scheduling model considering the uncertainty of Price-Based Demand Response (PBDR) and incentive-based demand response (IBDR). In this model, level of risk affordability determines the output of the PV and power shortage punishment cost where the risk coefficient follows the risk attitudes of the policymakers. In a similar work, (Bai et al., 2016) proposes an interval optimization-based coordinated operating strategy for the gas-electricity Integrated Energy Systems. (Brandoni & Renzi, 2015) divides sizing constrains and related uncertainty to five groups: i) energy prices, ii) ambient conditions, iii) energy demand, iv) units' characteristics, and v) electricity grid constraints. The hybrid renewable system under this analysis is made up of an electrical solar device and a micro-Combined Heat and Power, micro−CHP unit coupled to a cooling device. (Parhoudeh, Baziar, Mazareie, & Kavousi-Fard, 2016) proposes a stochastic framework to handle the uncertainty effects in the optimal operation of micro grids. The fuzziness and randomness of qualitative parameters are included in form of a cloud model.

To evaluate the financial takeaway of Combined Cooling, Heating And Power (CCHP) systems, (Gu et al., 2014) presents an overall review of the modeling, planning and energy management of the CCHP microgrid. (Hu & Cho, 2014) Optimizes CCHP operation simultaneously with multiple objectives such as minimizing operational cost, Primary Energy Consumption and Carbon Dioxide Emissions considering the reliability of the CCHP operation strategy. (Fuentes-Cort, Vila-Hernndez, Serna-Gonzlez, & Ponce-Ortega, 2015) proposes an integrated system to provide hot water and electricity for domestic use in a residential complex. The model takes into account the time variation of the energy demands, different rates of purchase – sale of electricity and seasonal temperature changes over time. (Benam, Madani, Alavi, & Ehsan, 2015) presents a reliability-constrained optimization approach to determine the number and size of CHP system components, including CHP units, auxiliary boilers, and heat-storage tanks. (Mahani, Farzan, & Jafari, 2017) investigates the optimal arrangement of a storage network with respect to uncertainties in electricity demand and devise a rule-based control scheme for its operation. Moreover, (Gahrooei, Zhang, Ashuri, & Augenbroe, 2016) studies the type of staged investment of installing residential PV system that maximizes the long-term payoff and realize the optimal time to delay the investment (referred in this paper as ‘Waiting’)

Our work can be also compared by papers that approach the associated risks and uncertainties by considering the stochastic dominance among feasible DER portfolios. (Gollmer, Gotzes, Neise, & Schultz, 2007) applies Stochastic benchmarks in conjunction with concepts of stochastic dominance for modeling risk aversion in optimization of energy systems with dispersed generation. (Wu, Lau, Tsang, Qian, & Meng, 2014) proposes algorithms to determine the optimal exploitation of renewable energy as well as the associated energy scheduling decisions through the virtual price that took account of the volatility of renewable energy. the choice of renewable energy distributions follows the principle of first-order stochastic dominance. (Jung & Tyner, 2014) conducts cost-benefit analysis with electricity price, PV system degradation and Failure rate of the system as uncertain input variables to determine the economics of adopting solar PV systems in Indiana based on three policy instruments; Federal tax credits, Net metering and Financing with tax deduction for interest paid. And finally, (Shakouri, Lee, & Choi, 2015) applies the Mean–Variance Portfolio theory with the objective of maximizing the hourly electricity output of PV systems, minimizing the hourly volatility in electricity output and, optimizing the risk-adjusted performance of community-based PV investment.

Using real options approach for investment in the energy sector has also been demonstrated by many authors, including (F. Farzan, Mahani, Gharieh, & Jafari, 2015), (Siddiqui & Maribu, 2009), (Zeng, Klabjan, & Arinez, 2015), (Smith & McCardle, 1998), (Hlouskova, Kossmeier, Obersteiner, & Schnabl, 2005), (Chung-Li & Barz, 2002) and (Madlener, Kumbaroǧlu, & Ediger, 2005). In (Wickart & Madlener, 2007), authors develop an economic model that explains the decision-making problem under uncertainty of industrial firms operation. They also account for the risk and uncertainty inherent in volatile energy prices that can greatly affect the valuation of the investment project. (Tadeu et al., 2016) Investigates the influence of real options on investments for eighteen combinations of retrofit measures to ensure cost-optimality of DER systems in EU.

Yongma Moon in (Moon, 2014) proposes a model for optimal investment time on energy storage systems under uncertainty over future profits. The author applied real option theory to the proposed model to provide an optimal investment threshold. (Tooryan, Tafreshi, & Bathaee, 2013) presents a method based on particle swarm optimization for determining the optimal size of distributed energy resources in the micro-grid. In more recent papers, (Martínez Ceseña, Capuder, & Mancarella, 2016) analyses real option approach under three different planning philosophies: 1) not investing in a Distributed Micro Grid (DMG) system (do-nothing); 2) investing immediately in a DMG system based on a forecasts (best view), and 3) investing immediately in a DMG system based on the full tree and upgrading the system based on a rolling horizon (multistage). (M. M. Zhang, Zhou, & Zhou, 2016) proposes a real options model for evaluating renewable energy investment by considering uncertain factors such as CO2 price, non-renewable energy cost, investment cost and market price of electricity in China. Empirical results of the paper show that the current investment environment in China may not be able to attract immediate investment, while the development of carbon market helps advance the optimal investment time. (Shahnazari, McHugh, Maybee, & Whale, 2014) and (Reedman, Graham, & Coombes, 2006) study the optimal timing of one potential business response to carbon pricing introduced by Australian government based on an American-style option valuation method. same as our work, The method provides a decision criterion that informs the investor whether or not to delay the investment. (Boomsma & Linnerud, 2015) and (Krogh, Meade, & Fleten, 2012) uses a real options approach to demonstrate how investors in power projects respond to market and policy risks resulting from fluctuations in electricity and/or subsidy prices. The proposed analysis indicates that the prospects of termination in current support scheme will slow down investments if it is retroactively applied, but speeds up investments if it is not. (Tian et al., 2017) utilizes Real option to approach power generation under uncertainties from the perspective of power generation enterprises. Moreover, (Kashani, Ashuri, Shahandashti, & Lu, 2014), (Siddiqui, Marnay, & Wiser, 2007), (Zhu, 2012), (Siddiqui et al., 2007), (Reuter, Szolgayová, Fuss, & Obersteiner, 2012), and (Fertig, Heggedal, Doorman, & Apt, 2014) are examples of papers that consider discrete time in evaluation of real option as it is done in this paper. Similarly,(Zou, Wang, & Wen, 2017) applies discrete Real Option evaluation among multiple mutually exclusive options to assess loss upon an unfavorable unfolding of future uncertainties in their decision making process.

The work by Takashima et al. (Takashima, Siddiqui, & Nakada, 2012) is one of the closest to our work in the literature, where the authors address the problem of investment flexibility over timing, sizing, and technology choice of a power plant. They assume a single source of stochastic variation due to price of electricity and ignore randomness of demand and technology prices. They solve the problem sequentially by first computing optimal capacity based on option value, and then using common real option formulation to compute price threshold and investment timing. For the technology choice, they formulate net present value (NPV) over two alternatives of nuclear and gas fired plants, and find conditions under which one is superior to the other.

Section snippets

Characteristics of input parameters

We narrow down the problem of optimal sizing and timing into a portfolio of two generation assets, namely Micro gas-turbine (GT) generation and Photovoltaic (PV) Solar panels, and three stochastic sources of inputs; gas price, daily load profile, and price of PV technology. Binomial probability tree is assigned for PV technology capital cost to model its decreasing trend. The capital cost of GT systems is assumed to be constant over the planning horizon. Future gas prices are modeled by a mean

Optimal sizing decision trees

Each path of MC simulation can be represented by 9 features: annual average of natural gas price over planning horizon (GP1 to GP4), PV technology cost (PV1 to PV4) and the shape of average daily load profile. Behavior analysis of optimizer begins with investigating the significance of each factor in determination of optimal cluster. On that note, by considering the 9 features as the design matrix and the optimal clusters as the target variable, we apply Random Forest to identify the importance

Sensitivity analysis of investment rules

Under the nominal conditions for the stochastic input processes, four major likely clusters have emerged being the most likely far from the rest. We note that the four most likely clusters remain intact by increasing the number of sample paths from 10,000–20,000. Next we examine the robustness of these results under a number of varying circumstances. We run a design of experiment over the three most influential input parameters: gas price mean level, volatility, and load shape variation (Table

Conclusion and future works

The problem of optimal sizing and investment timing for a portfolio of gas-fired generation and PV is presented and solved using a computationally efficient approximation technique. The methodology takes advantage of input-output behavior of the portfolio that can be clustered with respect to the optimal capacity of its assets. The optimal time of investment is then calculated for optimal sizes using the real option approach. Furthermore, using RPART algorithm, decision trees are developed to

Declaration of Competing Interest

There is no conflict of interest.

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