Operations Research Helps the Optimal Bidding of Virtual Power Plants

Introduction

Distributed energy resources (DERs) that can replace traditional energy resources such as coal and nuclear energy continue to emerge. DERs refer to small and geographically dispersed technologies for generating and managing electricity that can decentralize the power grid. They include renewable energy (e.g., solar photovoltaics (PV) and wind), small-scale combined heat and power, energy storage systems (ESSs), and demand response (DR) (ACEEE 2021). DR serves as a DER by reducing electricity usage rather than generating actual electricity. According to their purpose and function, different types of DR can participate in electricity markets. For example, the New York Independent System Operator defines two types of DR: reliability-based DR and economic-based DR (NYISO 2020). The former aims to reduce usage instantaneously to secure the reliability of the power system, whereas the latter can participate in the day-ahead or auxiliary service markets to replace electricity generation from other resources. These DERs are growing rapidly worldwide. According to the International Renewable Energy Agency, the share of renewable energy in the power sector is expected to increase from 25% to 85% by 2050, mostly through growth in solar and wind power energy (IRENA 2019). Furthermore, the capacity of global DR markets is predicted to increase from 39 GW in 2016 to 144 GW by 2025 (Guidehouse-Insight 2016). Thus, DERs are expected to play increasingly vital roles in the future electricity markets. However, the characteristics of DERs such as their unstable and relatively low output raise their barriers to entry for participation in the power market.

To overcome these barriers to market entry, the virtual power plant (VPP) concept was introduced in the power market in the mid-2000s. Although several definitions exist, the initial concept was established by a European project called the Flexible Electricity Networks to Integrate the Expected Energy Evolution project (Pudjianto et al. 2007). This project defined the VPP concept as follows: “A VPP aggregates the capacity of many diverse DERs. It creates a single operating profile from a composite of the parameters characterizing each DER and can incorporate the impact of the network on aggregate DER output. A VPP is a flexible representation of a portfolio of DERs that can be used to make contracts in the wholesale market and offer services to the system operator.” That is, a VPP is a cloud-based IT platform that collects several small DERs and operates as a single power plant in the electricity market. It can control all individuals and manage the electrical energy flow between them to ensure a reliable and efficient power supply to power consumers. Figure 1 illustrates the VPP concept.

VPPs can be categorized into supply-based, demand-based, and mixed VPPs, depending on their resource composition (Guidehouse-Insight 2020). First, supply-based VPPs collect only distributed power generation resources such as renewable energy power plants. As conventional power plants provide power into the electricity market, supply-based VPPs also deliver real electricity generated from small power plants to the market. Demand-based VPPs collect only DR resources from some end-use power consumers by inducing them to reduce their electricity usage. This type of VPP is called a DR aggregator or curtailment service provider. They do not supply power to the market; rather, the market considers the demand curtailment as the power supply. Finally, mixed VPPs, which are considered the ultimate goal of a VPP, incorporate both distributed power generation and DR resources. The various types of VPPs must focus on different purposes to operate their platform efficiently. A supply-based VPP identifies the generation pattern of renewable energy power plants and manages controllable power plants on its platform. By contrast, a demand-based VPP calculates the demand pattern of end-use consumers on its platform and finds a way to cut (or shift) some of their consumption. Meanwhile, a mixed VPP should have the capabilities of both supply-based and demand-based VPPs.

As advanced metering infrastructure, control, and sensor technologies were developed in the early 2010s, the application of DR programs and markets gradually expanded in scope. Correspondingly, demand-based VPPs emerged worldwide, including PJM in the United States, the United Kingdom, Australia, South Korea, and Japan (Bertoldi et al. 2016). Furthermore, markets for supply-based VPP trading have recently opened in many countries. For example, with the rapid expansion of renewable energy in California, the California Independent System Operator (CAISO) is promoting DER participation in the power market. The CAISO has launched a distributed energy resource provider to participate in the CAISO power market by integrating a DER capacity of 0.5 MW (CAISO 2016). In Australia, a brokerage system has been introduced to lower the barriers to entry into the power market for small DERs. As a result, the so-called small generation aggregator is recruiting small power generation resources of less than 30 MW and selling them to the power market (AEMO 2020). Furthermore, according to recent reports, the markets for VPP are expected to shift toward the mixed VPP model globally (Guidehouse-Insight 2019, Allied Market Research 2020). Therefore, current markets for demand-based and supply-based VPPs are expected to merge in the future.

Need for Analytics

Similar to other platform businesses such as Uber and Airbnb, the competitiveness of VPPs depends on their data analytics and operational technologies. First, a VPP connects unstable and volatile DERs and monitors circumstances on the platform and in the electricity market. Second, the VPP requires high-performance data-processing capabilities and intelligent forecasting. Third, the VPP controls DERs to balance supply and demand on the VPP platform as well as the bid and actual supply amounts. Finally, the VPP should provide additional value to the stakeholders that own DERs and participate on the platform.

In this study, among the several operational challenges of VPPs, we focus on the optimal bidding strategy—that is, the amount of power the VPP commits to supply. A VPP generally participates in the day-ahead electricity market without perfect information on the power supply from DERs, particularly renewable energy. Renewable energy and flexible demand can cause violent fluctuations in the power supply. In addition, their uncertainty significantly affects the management objectives of VPPs such as revenue maximization and cost minimization (Yu et al. 2019). In this context, it is challenging to ensure that the actual supply is close to the bid amount (Liang and Guo 2016). When a VPP submits overly conservative bid amounts, it can forfeit the compensation from the additional electricity supply to the market. On the contrary, if it commits to excessive bid amounts, it can incur significant penalties when the amount of electricity supplied from DERs falls short of expectations. Therefore, the bidding strategy can directly determine a VPP’s profits and affect the competitiveness of the company in the new market.

Literature Review

The stream of the OR/MS literature on market mechanism design in new platform businesses is rapidly growing. In particular, several studies focus on how to encourage stakeholders to participate on platforms. Cachon et al. (2017) analyze the pricing design of self-scheduling platforms such as Uber. Guda and Subramanian (2019) investigate on-demand workers’ strategies on a platform and propose a more effective incentive payment mechanism. Vinsensius et al. (2020) propose an incentive mechanism for an e-commerce–attended home delivery platform to maximize its overall profit. Other studies focus on analyzing the effects of incentive mechanisms in platforms. Burtch et al. (2018) explore the extent to which financial incentives stimulate online reviews on platforms such as Amazon Mechanical Turk. Kuang et al. (2019) examine the influence of incentives on activities on knowledge exchange platforms. Liu et al. (2021) find that incentives can reduce the moral hazard of an Uber driver by comparing taxis with the price proportional to the travel time. Many of the studies related to platform businesses have focused on incentive-based mechanisms similar to the problem in our study.

More specific to VPP platforms, early studies focused on the optimal bidding strategies of VPPs over the past decade because their strategy plays a crucial role in maximizing revenue. Mashhour and Moghaddas-Tafreshi (2010) address the optimal bidding strategy when participating in energy and spinning reserve markets based on deterministic price-based unit commitment. Vasirani et al. (2013) develop a bidding strategy by applying deterministic optimization methods. However, these studies have a limitation whereby DER uncertainty is not taken into account, meaning that the optimal bidding strategy of a VPP cannot adequately respond to sudden changes in power supply and demand. Consequently, it can fail to meet its power supply commitment, which can reduce the new power plant’s profits drastically.

To overcome the limitations of studies that ignore uncertainty, the most recent studies consider the uncertainty in deriving the optimal bidding strategies of VPPs. These studies focus on extending deterministic optimization models to stochastic optimization models. Many aim to quantify the uncertainty arising from power generation and demand in VPPs and use these quantifications to optimize VPP operations. For example, Shi et al. (2014) and Fazlalipour et al. (2019) quantify the uncertainty of power generation and demand of VPPs by applying Latin hypercube sampling to generate scenarios. Zamani et al. (2016) generate scenario trees by applying point estimation methods to quantify the uncertainty in power prices and power generation. Beraldi et al. (2018) adopt the Monte Carlo technique to generate power generation, demand, and electricity price scenarios. On the contrary, Baringo et al. (2018) and Tang and Yang (2019) present a robust optimization method for quantifying uncertainty.

In stochastic programming problems, a two-stage stochastic optimization approach is frequently used. This tendency reflects the prevalence of the day-ahead bidding process in the electricity market. Ju et al. (2016) are the first to introduce the general approach. They consider a day-ahead electricity market in which the day-ahead bids are regarded as here-and-now decisions, whereas decisions on the day of operation are represented as second-stage wait-and-see (WS) decisions under which uncertainty is resolved. Fazlalipour et al. (2019) construct a revised model determining the optimal bids for both day-ahead energy and reserve markets in the first stage and schedule in the second stage. Similarly, Beraldi et al. (2018) determine the procurement plan through the available conventional production units, bilateral contracts, and trading in the day-ahead market in the first stage. By contrast, storage units and the energy traded in balancing markets are decided in the second stage.

However, a two-stage stochastic programming model has some crucial limitations. First, the model determines the optimal bidding strategy based on the assumption that decision makers know all the information on the day of operation at the beginning of the second stage. Because the power generation from most DERs remains uncertain until the day of operation, this approach lacks practicality. Second, in the two-stage stochastic programming model, it is impossible to consider situations in which the forecasted DER amounts are updated. However, forecasts continue to be updated throughout the day of operation, and the decision maker also tries to use these forecasts. Some recent studies note the limitations of two-stage models and adopt a multistage framework in the generalized hypothetical market environment (Ottesen et al. 2018) and Spanish day-ahead and intraday markets (Wozabal and Rameseder 2020). These studies exploit the multistage approach that can obtain more flexible decisions for the realizations of uncertain parameters (Huang and Ahmed 2009). In these studies, decision makers thus modify their decisions according to the realization and update of DER forecasts on the day of operation. Therefore, we also use the multistage approach in this study.

Contributions

In this work, we collaborate with a company, H Energy Co. Ltd., which participates in the DER market in South Korea, and use the data provided by the company. Moreover, we incorporate the following points to develop a model and a solution approach that is practically applicable in the real world. The contributions of this work are summarized as follows.

First, unlike most previous studies that rely on two-stage stochastic programming models, we use the multistage stochastic optimization approach. In this model, uncertainty still exists during intraday operations, and solar PV generation and demand forecasts are updated every three hours with updated weather forecasts. Our model also differs from those of previous studies that use the multistage approach (Ottesen et al. 2018, Wozabal and Rameseder 2020). We consider ESSs as the sources of DR resources and optimize the timing of the balance (or imbalance) between the bid amount and actual supply. These differences make our model more realistic.

Second, to generate a scenario lattice for our multistage approach, we devise a framework that considers the characteristics of a recently proposed machine learning–based electricity demand forecasting model. H Energy Co. Ltd. uses a state-of-the-art clustering technique to forecast the electricity demand of prosumers in its VPP. Therefore, we cannot use conventional scenario-generation methods. We thus propose a method that considers the company’s situation. In our method, we take advantage of updated data to generate a scenario lattice on a daily basis.

Third, this is the first study to derive an optimal bidding strategy under an incentive-based market structure, not the kind of penalty-based market structure considered in other studies. Both structures induce VPPs to reduce the difference between the bid amount and actual supply. The penalty proportionally increases with the difference. Hence, previous studies have linear objective functions in their optimization models. By contrast, the South Korean market offers all-or-nothing incentives based on a certain threshold difference between the bid amount and actual supply. This incentive-based market structure complicates our optimization model. Consequently, we believe that we solve the newly proposed problem for the first time. We also claim that this study provides unique insights for other countries and regions that aim to introduce DER markets by considering the market structure.

Finally, we design heuristic techniques to reduce the computational burden in the optimal solution-finding procedure. For the company to use our proposed model in practice, the computational time required to obtain the optimal solution should be reasonable. However, finding the optimal solution is time-consuming. In addition, when considering the incentive-based market structure, the complexity of the optimization model formulated in this study increases drastically. Our heuristic techniques can resolve the problem of extended computational time.

Problem Description

In South Korea, a DR trading market for demand-based VPPs has been in operation since November 2014. The market rules for the DR trading market are similar to those in overseas markets (Daniels and Lobel 2014, Nekouei et al. 2014, Campaigne and Oren 2016, Henríquez et al. 2017). The market system for supply-based VPPs opened in February 2019 after years of discussion. The system’s market structure is similar to those of overseas markets. However, to promote the market in its development stage, the South Korean government incentivizes VPPs to meet their bid amounts rather than imposing penalties, which occurs in some overseas markets. The specific rules for the new market system for supply-based VPPs set in 2020 are summarized here:

1. Each DER in the VPP should be renewable energy, ESS, or electric vehicles with a capacity below 1 MW.

2. The VPP participates in the day-ahead wholesale electricity market in which each participant submits two bidding profiles at 10 a.m. and 5 p.m. on the day before operation and the average value of the two bids becomes the final bid amount.

3. The VPP can qualify as a market participant only if it maintains an average error rate between the final bid and actual supply amounts of less than 10% during a test period.

4. A balancing incentive of 3 KRW (Korean won) per kilowatt-hour applies when the bidding error for each hour is below 8% and the supply amount exceeds 10% of the DERs’ total capacity in the VPP.

In summary, two separate DER markets, demand-based VPPs and supply-based VPPs, now exist in South Korea. However, consistent with the global trend, the South Korean government is planning to merge them into a market for a mixed-type VPP. Given this development, we consider the situation in which these two markets are integrated so that mixed-type VPPs can participate in the DER market.

Definition of VPPs

In this study, we consider a mixed-type VPP that has only solar PV and DR resources, as described in Figure 2. Solar PV generators only interact with the VPP platform, and their aggregated electricity is sold in the DER market via the VPP. Meanwhile, large commercial buildings and/or factories with ESSs obtain electricity from a utility company and pay for their usage (demand). The VPP obtains only DR resources by operating their ESSs, and brokers direct the resources to the DER market. The DR resources are obtained only when the net demand of each large commercial building or factory is below the customer baseline load (CBL), as in the current DR trading market in South Korea. The CBL is determined as the moving average of its demand over the previous five days. When the VPP operates the ESS to maximize the platform’s profit, it considers both solar PV generation and bid amount. In this situation, buildings and/or factories sometimes need to deviate from the optimal ESS operation to minimize their own electricity bills. Although the deviation can maximize the platform’s profit, buildings and/or factories can be worse off in terms of their payments for electricity usage (demand). However, the VPP can compensate for their profit reduction by using additional balancing incentives derived from the deviations.

Decision of Optimal Bidding Profile

Figure 3 illustrates the flow of information updates and decisions for the VPP. The VPP first updates the DER forecasts and determines the bidding profile in the day-ahead market. The bidding profile includes the electricity amount of DERs that the VPP can commit to supply to the market for each hour on the day of operation. For a given bidding profile, the VPP operates controllable DERs (i.e., ESS) on the platform to balance the actual DER supply and bidding profile on the day of operation. The VPP’s revenue is realized depending on the balancing results.

In this study, we aim to find the optimal bidding profile for the VPP in the day-ahead market. We assume that the VPP can focus on the profile from 6 a.m. to 6 p.m. because it only has solar PV and DR resources. Solar PV only generates electricity during the day, and DR resources cannot be sold when the system marginal price (SMP) of electricity is below a certain threshold (i.e., demand in the electricity market is sufficiently low at night). By assumption, the decision outputs in the day-ahead bidding stage are the optimal bidding profiles for the period from 6 a.m. to 6 p.m., when the amounts potentially exceed 10% of the DER’s capacity. We assume that the VPP is uncertain about the solar PV generation amount and demand of large commercial buildings and/or factories (DR amounts) on the day of operation. However, following Shi et al. (2014) and Fazlalipour et al. (2019), we assume that the electricity price (i.e., the SMP) can be perfectly forecasted because its volatility is relatively low in the South Korean market and daily variation is negligible. Further analysis with real data in the Pilot Test section justifies this assumption. In addition, similar to most previous studies (Mashhour and Moghaddas-Tafreshi 2010, Zhao et al. 2015, Zhang et al. 2016, Henríquez et al. 2017, Wang et al. 2017, Baringo et al. 2018, Wozabal and Rameseder 2020), we assume that the VPP acts as a price-taker. In other words, its bidding decision does not affect the SMP, as the electricity market in South Korea is dominated by large conventional generators. The capacity of the 16 major generators ranges from 1,200 to 28,600 MW, and these account for 78.3% of the market (KPX 2021a). By contrast, although more than 50 VPPs participate in the market, their contribution is less than 0.003% (KPX 2021b).

For intraday operations, we assume that all the ESSs on the platform are fully charged at the beginning of the period. In addition, we consider the situation in which weather information is updated every three hours in South Korea. Therefore, the VPP updates the solar PV generation forecasts, demand forecasts, and the platform’s operation depending on the latest weather information. Hence, each decision point for ESS operation scheduling during intraday operations occurs immediately after the DER forecasts are updated, as presented in Figure 3.

Formulating the Problem

We formulate the previously described multistage decision problem in the form of a finite-horizon discrete-time Markov decision process (MDP) model. Appendix B presents the details of the model, which is composed of five stages. The first stage represents the time when the VPP submits the bid for the day-ahead market. In accordance with the forecast-updating process, we divide the 12-hour period on the day of operation into four periods. Thus, all the remaining stages are defined as the beginning of the three-hour intervals on the day of operation.

Naturally, the state space, action space, and immediate reward structures differ between the first and other stages. In the first stage, the state variables are the solar PV generation and electricity demand forecasts for the day of operation, and the decision variable is the bidding profile. The immediate reward is defined as the VPP’s revenue from DER sales as the bid amounts. Meanwhile, in the remaining stages, the state variable consists of the initial state of charge in the ESS, solar PV amounts generated, and electricity demand in the stage. Based on these variables, the model determines the amount of charge/discharge during the stage. Then, the immediate rewards are the operating costs of charging/discharging the ESS, as per the decision variables. All the state and decision variables for the first stage make up a sequence of 12-hour values. However, those for each of the remaining stages make up a sequence of three-hour values.

The objective function, which represents the total expected profit of the VPP, includes its revenue from the sale of DERs as the bid amounts, its expected costs, and balancing incentives during the intraday operations. The model has several constraints for intraday operations, such as the conventional operational constraints of ESSs. In addition, several constraints are introduced to determine whether balancing incentives are available.

Solution Approach

To solve the MDP model described in the previous section, we use the simple stochastic dynamic programming approach. This approach incorporates the uncertainty in the DER amount forecasts into the model using a scenario tree or lattice. We first quantify the uncertainty and then generate the tree for the solar PV amounts based on a conventional method and the lattice for the DR amounts based on a newly devised method.

Dynamic Programming Approach

In our MDP model, the value of being in a certain state with optimal decisions in every stage can be defined using the Bellman optimality equation. Each value function in the operation stage represents the optimal expected cost during the remaining intraday operation time given the bidding profile. Meanwhile, the value function in the first stage represents the optimal total expected profit from selling DERs in the market. The corresponding decision in the first stage is the optimal bidding profile. To derive these optimal values and the decision, we use the standard backward-induction approach. Although advanced methodologies for the solution approach have been developed, these are complex for practitioners to understand and require excessively strong assumptions. For example, the stochastic dual dynamic programming approach used by (Wozabal and Rameseder 2020) requires an assumption about the stage-wise independence of the uncertainty. The assumption is too strong for the methodology to be applied to our problem. Owing to these practical issues, we use the simplest approach.

Uncertainty Description

To describe the uncertainty in the DER forecast amounts in our proposed model, we use two approaches. The forecast error can be well defined when a stochastic time-series model is used to forecast DERs. Hence, in this case, we use the well-developed conventional uncertainty quantification and scenario tree generation methods. However, as H Energy Co. Ltd. continues to research forecasting models, we cannot always use conventional methods. That is, if the DERs are forecasted based on recent machine learning–based models, it is difficult to quantify their forecast error. Then, new methods are required to describe their uncertainty. Therefore, for each case, we attempt to establish two frameworks that the company can apply to future forecasting models. We use the conventional framework for solar PV and devise a new framework for electricity demand (DR amounts).

First, we use the conventional framework to describe the uncertainty in the solar PV generation forecast, as illustrated in Figure 4(a). To describe the conventional framework simply, we take the seasonal autoregressive integrated moving average (SARIMA) model, which is the simplest model for seasonal time-series forecasting (Box et al. 2015). Using the solar PV generation data, we estimate the SARIMA model and fit the distribution of its error term. Based on the distribution, we quantify the uncertainty in the solar PV forecast and use the moment-matching method to generate the scenario tree (Høyland and Wallace 2001, Høyland et al. 2003).

Second, and more importantly, we devise a new framework to describe the uncertainty in the electricity demand forecast. The company first investigates data on prosumers’ demand. Similar to several recent studies (Aghabozorgi et al. 2015, Teeraratkul et al. 2017), the company finds that the electricity consumption pattern of prosumers is similar over preceding days. Consequently, H Energy Co. Ltd. develops a clustering-based model to forecast the prosumers’ electricity demand. During the forecasting procedure, the model shows several clusters of demand patterns for each period and the transition probabilities between clusters in different periods. In addition, by using the data from previous days, the model selects some clusters as possible patterns for the next day. Based on these selected clusters and the transition probabilities between them, we generate a scenario lattice for the uncertainty in the electricity demand forecast. The lattice is updated daily, as the clusters of possible patterns for the next day are selected based on data on recent days. Figure 4(b) illustrates the framework, and Appendix C summarizes its detailed procedure. Through empirical tests, we verify that our scenario lattice explains the uncertainty in the electricity demand forecasts well. We also show that the scenario lattice has lower computational complexity than the scenario tree.

Implementation

To verify the performance of our method, including our formulated model and proposed solution approach, we first evaluate the performance in the pilot test under a simple simulation environment. We define the simulation environment in collaboration with H Energy Co. Ltd. based on DERs collected by the company. Additionally, the company implements the method practically as a component of its platform.

Pilot Test

For the pilot test, we assume that the VPP has 27 solar PV generators and one DR resource on the platform. As illustrated in Figure 5, the 27 solar PV generators are mostly distributed in Chungcheongbuk-do in South Korea, and their aggregated capacity is 1,354.5 kW. As the DR resource, a commercial building with an ESS in Seoul participates on the platform. This ESS has a storage capacity of 175 kWh and a charging/discharging capacity of 100 kW. The state of charges can vary from 10% to 90% of the storage capacity to retain battery life. The data for solar PV generation and the electricity demand of the building are extracted from the database of H Energy Co. Ltd. In addition, we obtain the data on the electricity market price, the SMP, from the Korea Power Exchange, and the data on the electricity usage rate for the building from the Korea Electric Power Corporation.

We focus on weekdays in summer and winter when the balancing incentive is substantial. After fitting the forecasting models using the data on January and February 2019 for winter and data on July and August 2019 for summer, we first perform the descriptive analysis of the uncertainty in the DER forecast amounts. Appendix D describes the estimated forms of forecasting models for solar PV generation. The standard deviation (SD) of the solar PV generation forecast error is about 6% of total capacity and about 20% of average generation amount. Meanwhile, the SD of the demand forecast error ranges from 15% to 25% of the average demand. Tables 1 and 2 present the analysis results. When we use the SMP on the day before operation as the forecast value of the SMP on the day of operation, the SD of the forecast error of the electricity price is only about 6.4% of the average SMP in the same period. Hence, the SD is much less than those for the previous DER amount forecasts. The analysis results justify our aforementioned assumption that we ignore the uncertainty in the SMP forecast.

Table 1. Characteristics of the Solar PV Generation Forecast Error Show the Form of Uncertainty in the Solar PV Generation Forecast

Table 2. Characteristics of the Electricity Demand Forecast Error Show the Form of Uncertainty in the Electricity Demand Forecast

The frameworks in Figure 4 quantify the uncertainty in the DER forecast amounts and generate corresponding scenario trees or lattices. Figures 6 and 7 illustrate the scenario tree for the solar PV generation forecast error in summer and a sample scenario lattice for the electricity demand forecast on a certain day. Based on the forecasting models and their uncertainty descriptions, we test the performance of our method for the test periods of January 2020 and August 2020.

For a given day in the test periods, our method derives the optimal bidding profile based on the forecasts. In addition, the intraday operations, which are the optimal ESS charging/discharging operations, and balancing results are determined given the profile and realized solar PV generation and electricity usage data. Then, we evaluate the balancing incentives and profits for the day based on the intraday operations. By analyzing the relative error of the solar PV forecasts during the training periods, we determine that our method only considers from 80% to 120% of the solar PV forecast as candidate bidding profiles to alleviate computational complexity. Moreover, we discretize the range into five levels in increments of 10%. Consequently, five bid candidates are generated in each stage. As each day is composed of four stages, the number of candidates for the bidding profile is 625.

To define the performance measure, we introduce comparing values of two solutions. The first is the profit of the solution derived when we have perfect information about the PV generation amount and electricity usage. The second is the profit of the solution derived when we only consider the DER amount forecast values without any description of the uncertainty. The former is similar to the value of the “wait-and-see” solution, and the latter is similar to the value of the “EEV (expected result of using the expected value problem)” solution. The performance measure is defined as the following ratio: the benefit of our method compared with the latter comparing value over the difference between the former and latter comparing values. This measure describes the relative position of our method between the two comparing values. That is, the measure’s value can lie between zero and one, and the measure is close to one when our model performs similarly to the model with perfect information. Table 3 summarizes the values of the performance measures in the pilot test.

Table 3. Performance of the Proposed Approach in the Pilot Test

Figure 8 describes an example day to show in detail how our method affects the performance measure. Three figures show the intraday operations and balancing results on a specific day under the optimal bidding profiles derived from our method and the two comparing solutions. The difference in bidding profiles causes the difference in incentive profits. Figure 8(a) illustrates that when the bidding profile is determined with perfect DER amount information, the company has a bidding incentive for the entire operation time. Meanwhile, as shown in Figure 8, (b) and (c), without perfect DER information, it is difficult to obtain balancing incentives during the entire operation time. However, different bidding profiles lead to different profits. Under the bidding profile derived from our method, the intraday operations miss the incentive at 10 a.m. By contrast, under the bidding profile derived for the latter comparing solution, in which we only consider the forecast of DER amounts without describing any uncertainty, the intraday operations miss the incentive at 11 a.m. Because the amount of DERs at 11 a.m. is larger than the amount of DERs at 10 a.m., the incentive at 11 a.m. is higher than that at 10 a.m. Consequently, the bidding profile derived from our method provides a higher profit.

Heuristics

Like most stochastic programming models, our method is affected by the curse of dimensionality. To alleviate this issue, we apply three heuristics to our method. The first heuristic is the conventional scenario reduction method (Dupacová et al. 2000, Heitsch and Römisch 2009). Using this method, we can reduce the calculation without any loss of information about uncertainty. Second, we use the tightening big-M method. Tightening big-M in integer programming is a popular method for reducing the running time (Camm et al. 1990, Morales-España et al. 2016). As introduced in Appendix E, we set and tighten several big-Ms tailored to each variable that requires a big-M parameter rather than the representative big-M parameter for all the variables. This method finds the minimum value of each big-M under which the corresponding constraint can never be violated. Therefore, this allows us to tighten the search space, and the computational burden decreases without a loss of solution quality. The last heuristic is the rule-based fixing value of the binary variables for determining whether the DR amount occurs at the time. We can fix the value of the binary variables for the occurrence of DR to one beforehand in a peak load period if the CBL is above the electricity usage forecast at that time. In addition, we can also fix the value of the variables in a light load period to zero if the CBL is below the electricity usage forecast at that time. Appendix F describes the details of the last heuristic. This heuristic method also reduces the search space by eliminating unnecessary branching operations (Maranas and Zomorrodi 2016). This method fixes the values of some of the binary variables, as the optimal solution is clearly based on a logical rule. Thus, it does not diminish the accuracy of the optimal solution.

By applying the above heuristics, the computational time required to select one optimal bidding profile is, on average, approximately 21 minutes in the pilot test, using a PC with an i7-7820X CPU, 80 GB RAM, and a solver GLPK in the Pyomo Python library. Our heuristic methods thus decrease the computational time by 14%, on average, from about 25 to 21 minutes. The general concern about applying these heuristics is the trade-off between solution quality and running time. However, as mentioned previously, we find that the heuristics do not lower solution quality.

Benefits and Use in Practice

The company develops a platform that operates the VPP business model. Figure 9 describes the platform structure. Various algorithms are developed and operated on the platform (colored in white in the figure). The company adds our method on the platform and uses it for the day-ahead DER bidding process, which is the most important algorithm for the platform’s profit.

Currently, the company continually attempts to advance the forecasting models for the DER amount. With the company’s latest forecasting model and large DERs for use in practice, our method shows the performance value of about 45%. Equipped with heuristics, the method selects the best bidding profile with a sharply decreased running time. Furthermore, the company can easily execute the method in the market. Thus, it can help the company raise its profits in the market.

Furthermore, as we develop our method that can be understood by the company, the company expects it to become flexible for use with other types of forecasting models and under various market regulations. In practice, the company is continually improving the accuracy of its forecasting model for solar PV. In addition, the VPP market’s regulations are expected to be continually modified until the market stabilizes. When a new forecasting model emerges or the VPP market’s regulations are revised, the company could easily modify the method in response to such changes.

Conclusions

The competitiveness of platform businesses depends on data analytics and operational technologies. From the OR/MS perspective, we proposed a method for the efficient operation of a new platform business in the electricity market. Among the several operational problems, we focused on how a VPP company determines the optimal bidding profile in the day-ahead electricity market. Through a two-year collaboration with a real-world company, H Energy Co. Ltd., we formulated the problem into a multistage stochastic programming model and proposed a new approach. Based on data analytics, we devised frameworks that describe the uncertainty in the model. Furthermore, we incorporated three heuristic techniques that reduce the computational burden. After verifying the performance of our method in a pilot test with real data, the company implements our method on its platform. It expects to adopt our method as newly developed forecasting models and changed market structures emerge.

We acknowledge some of the limitations of our study and suggest potential extensions for future research. First, we assumed that the VPP acts as a price-taker. However, as more DERs and accordingly more VPPs penetrate the electricity sector in the future, the actions of VPPs will affect prices. Hence, a natural extension to our model would be to incorporate a price-supply function (Ding et al. 2017). Furthermore, the model must consider how the interactions among the strategic behaviors of several VPPs change the market (Kazempour and Zareipour 2014). Second, our method considered fixed candidates for the bidding profile. As abundant data are collected to analyze the bidding profile and relevant information, we can eliminate beforehand the impractical bidding profiles by incorporating machine learning techniques such as multidimensional classification (Read et al. 2014, Jia and Zhang 2020). This would improve the practicality of our method. Despite these limitations, our proposed method can readily be applied in the South Korean VPP market. In addition, it can be used by VPP companies overseas because their market structures resemble those in South Korea. As the DER markets expand and VPP companies emerge, our study can serve as a key reference for VPP companies interested in improving the efficiency of their services.

Appendix A. Table of Abbreviations

To help readers understand our study better, we summarize the definitions of abbreviations used in the study in Table A.1.

Table A.1. Definitions of Abbreviations in the Study Are Summarized

Table A.1. Definitions of Abbreviations in the Study Are Summarized

Abbreviation	Definition
ARIMA	Autoregressive integrated moving average
CAISO	California Independent System Operator
CBL	Customer baseline load
DER	Distributed energy resource
DR	Demand response
ESS	Energy storage system
KRW	Korean won
MDP	Markov decision process
PV	Photovoltaics
SARIMA	Seasonal autoregressive integrated moving average
SD	Standard deviation
SMP	System marginal price

Appendix B. Mathematical Formulation

Sets and Indexes

• $j\in J$: Set of solar PV generators.

• $c\in C$: Set of companies with ESS.

• $t\in T=\{1,2,3,4,5\}$: Set of stage parameters; $T^{\prime }=\{2,3,4,5\}\subset T$

• $h\in H_t=\{3t,3t+1,3t+2\}$: Set of three-hour time for each stage $t\in T^{\prime }$.

In this multistage setting, the first time period, t = 1, presents the time when the VPP submits the bid for the day-ahead market. Conversely, all other remaining periods in $T^{\prime }$ are defined as the beginning of the three-hour intervals during the operation day. Each hour h represents the time interval $[h,h+1)$. Therefore, for example, H₂ represents three hours from 6 a.m. to just before 9 a.m.

Parameters

• ${\lambda }_{\mathit{SMP}}(h)$: SMP at time h (e.g., KRW/kWh).

• ${\lambda }_{\mathit{elec}}(h)$: Electricity price for usage of companies at h (e.g., KRW/kWh)

• ${\lambda }_{\mathit{inc}}(h)$: Balancing incentive at time h (3 KRW/kWh in the Korean market).

• $\overline{d^c(h)}$: Moving average of $d^c(h)$ for the previous five days, which is defined as CBL in this study.

• $E_{\mathit{min}}^c,\hspace{0.17em}{E}_{\mathit{max}}^c$: Minimum, maximum ESS storage capacity of company c (e.g., kWh).

• $B_{\mathit{max}}^c$: ESS charging and discharging capacity of company c (e.g., kW).

• ν_c,ν_d: Charging, discharging efficiency of ESS (e.g., %).

• β: Indicator for tax combined fare ($\beta =1.137$ in the Korean market).

• α: Incentive coverage denominator (the incentive coverage is 8% in the Korean market, so $\alpha =0.08$ in this study).

• ρ: Total capacity of all PV generators in the VPP platform (e.g., kW).

Parameters are mainly defined for electricity and DER markets and the constraints for ESS. Some parameters such as tax β, balancing incentive price ${\lambda }_{\mathit{inc}}(h)$, and incentive coverage α are specified by the rules in the Korean market.

State Variables

• $Z^j(h)$: A day-ahead forecast amount of generation by the generator j at time h (e.g., kWh).

• $△Z^j(h)$: Forecast error of generation by generator j in error at time h (e.g., kWh).

• $G^c(h)$: A day-ahead forecast amount of consumption by company c at time h (e.g., kWh).

• $△G^c(h)$: Forecast error of power consumption by company c at time h (e.g., kWh).

• $e^c(h)$: ESS state of charge at h of company c (e.g., kWh).

• $e^c(5)$: Initial ESS state of charge of company c (e.g., kWh).

• ${\mathbf{s}}_1={\{({\sum }_{j\in J}{Z}^j(h)))}_{h\in H_t,t\in T^{\prime }},(G^c(h)){)}_{c\in C,h\in H_t,t\in T^{\prime }}\}$: State in the stage t = 1.

• ${\mathbf{s}}_t=\{({\sum }_{j\in J}(Z^j(h)+△Z^j(h)),{(G^c(h)+△G^c(h))}_{c\in C}{)}_{h\in H_t},{(e^c(3t-1))}_{c\in C}\}$: State in the stage $t\in T^{\prime }$.

For the first stage, state variables are the forecasts of solar PV generation and electricity demand for all time periods of the other stages in $T^{\prime }$. We set that each company’s ESS is fully charged at the beginning of the operation day because the electricity price at nighttime is low. That is, $e^c(5)=E_{\mathit{max}}^c$. Conversely, for the remaining stages, the state space is defined by three variables: the state of charge in the ESS at the beginning of the stage, the solar PV amounts generated, and the electricity demands in the stage.

Decision Variables

• q(h): Bidding amount at time h (e.g., kWh).

• $b_{\omega }^{c,ch}(h)$: Battery charging power at time h by company c in scenario ω (e.g., kW).

• $b_{\omega }^{c,dc}(h)$: Battery discharging power at time h by company c in scenario ω (e.g., kW).

• $I^{+}(h),I^{-}(h)$: Binary variables for checking the balancing incentive at time h.

• ${\mathbf{a}}_1=\left\{{(q(h))}_{h\in H_t,t\in T^{\prime }}\right\}$: Action in the stage t = 1.

• ${\mathbf{a}}_t=\{{(b^{c,ch}(h),b^{c,dc}(h))}_{c\in C,h\in H_t},{(I^{+}(h),I^{-}(h))}_{h\in H_t}\}\hspace{1em}\forall t\in T^{\prime }$: Action in the stage $t\in T^{\prime }$.

As in the case of the state, we need to define the actions differently for the first stage and the other stages. For the first stage, action variable is the bidding profile for all time periods of the other stages in $T^{\prime }$. In contrast, for the other stages, action variables are the ESS charging and discharging amounts during each hour and binary variables to identify whether the VPP can obtain the balancing incentive or not at each hour.

To specify feasible values of decision variables, the VPP needs to consider several constraints for the intraday operation as follows:

$$d^c(h)=G^c(h)+△G^c(h)-b^{c,dc}(h)+b^{c,ch}(h)\hspace{1em}\hspace{1em}\forall c\in C\hspace{0.17em},\forall h\in H_t,$$(B.1)

$$-B_{\mathit{max}}^c\le \frac{b^{c,dc}(h)-b^{c,ch}(h)}{△h}\le B_{\mathit{max}}^c\hspace{1em}\forall c\in C\hspace{0.17em},\forall h\in H_t,$$(B.2)

$$e^c(h)=e^c(h-1)+b^{c,ch}(h){\nu }_c-\frac{b^{c,dc}(h)}{{\nu }_d}\hspace{1em}\hspace{1em}\forall c\in C\hspace{0.17em},\forall h\in H_t,$$(B.3)

$$E_{\mathit{min}}^c\le e^c(h)\le E_{\mathit{max}}^c\forall c\in C\hspace{0.17em},\forall h\in H_t,$$(B.4)

$$q(h)-{\displaystyle \sum _{j\in J}\hspace{0.17em}}[Z^j(h)+△Z_{\omega }^j(h)]-{\displaystyle \sum _{c\in C}{v}^c}{(h)}^{+}\le \alpha \rho (1+M_1(h)*I^{+}(h))\forall h\in H_t,$$(B.5)

$${\displaystyle \sum _{j\in J}\hspace{0.17em}}[Z^j(h)+△Z_{\omega }^j(h)]+{\displaystyle \sum _{c\in C}{v}^c}{(h)}^{+}-q(h)\le \alpha \rho (1+M_1(h)*I^{-}(h))\hspace{0.17em}\forall h\in H_t,$$(B.6)

$$v^c{(h)}^{+}-v^c{(h)}^{-}=\overline{d^c(h)}-d^c(h)\hspace{0.17em},\forall c\in C\hspace{0.17em},\forall h\in H_t\hspace{0.17em},\forall t\in \hspace{0.17em}{T}^{\prime },$$(B.7)

$$0\le v^c{(h)}^{+}\le M_2(h)·v{i}_{\omega }^c{(h)}^{+}\hspace{0.17em},\hspace{1em}\hspace{1em}\forall c\in C\hspace{0.17em},\forall h\in H_t\hspace{0.17em},\forall t\in \hspace{0.17em}{T}^{\prime },$$(B.8)

$$0\le v^c{(h)}^{-}\le M_2(h)·v{i}_{\omega }^c{(h)}^{-}\hspace{0.17em},\hspace{1em}\hspace{1em}\forall c\in C\hspace{0.17em},\forall h\in H_t\hspace{0.17em},\forall t\in \hspace{0.17em}{T}^{\prime },$$(B.9)

$$v{i}^c{(h)}^{+}+v{i}^c{(h)}^{-}=1\hspace{0.17em},\forall c\in C\hspace{0.17em},\forall h\in H_t\hspace{0.17em},\forall t\in \hspace{0.17em}{T}^{\prime },$$(B.10)

$$v{i}^c{(h)}^{+},v{i}^c{(h)}^{-}\in \{0,1\}\hspace{0.17em},\forall c\in C\hspace{0.17em},\forall h\in H_t\hspace{0.17em},\forall t\in \hspace{0.17em}{T}^{\prime },$$(B.11)

• $v^c{(h)}^{+}$ and $v^c{(h)}^{-}$: Dummy variables for net DR amount at time h.

• $v{i}^c{(h)}^{+}$ and $v{i}^c{(h)}^{-}$: Dummy binary variables to indicate whether net DR amount exists or not at time h.

• $M_1(h)$: A large value for checking the balancing incentive.

• $M_2(h)$: A large value for the big-M linearization technique.

The first constraint (B.1) defines the net demand of each company. The constraints from Equations (B.2) to (B.4) represent the conventional operational constraints of ESS. The constraint (B.2) describes the upper bound of the net charging amount of ESS, which is determined by the PCS capacity of ESS. The constraint (B.3) represents the charging amount transition in the ESS, and the constraint (B.4) constrains the amount of stored electricity in the ESS. Constraints (B.5) and (B.6) are introduced to determine whether balancing incentives are available based on the comparison between the bidding and actual supply amounts. To determine the eligibility of the balancing inventive, we use two binary variables, $I^{+}(h)$ and $I^{-}(h)$, with an arbitrary large value $M_1(h)$. When the left-hand sides of Constraints (B.5) and (B.6) are smaller than $\alpha \rho$, both binary variables can become zero. This situation implies that the VPP can receive the balancing incentive at the hour. Here, $v^c{(h)}^{+}$ in the constraints (B.5) and (B.6) represents the DR amount that can be calculated as the positive amount of difference between the CBL and net demand. That is, when $\overline{d^c(h)}-d^c(h)$ is positive, $v^c{(h)}^{+}$ needs to the positive but $v^c{(h)}^{-}$ needs to become zero. In contrast, when $\overline{d^c(h)}-d^c(h)$ is negative, $v^c{(h)}^{+}$ needs to become zero but $v^c{(h)}^{-}$ needs to be the absolute of the negative value. To guarantee these situations, additional constraints from Equations (B.7) and (B.11) are required.

State Transition

• $e^c(3(t+1)-1)=e^c(3t-1)+{\sum }_{h\in H_t}[b^{c,ch}(h){\nu }_c-\frac{b^{c,dc}(h)}{{\nu }_d}]\forall c\in C,\forall t\in T^{\prime }.$

Basically, this state transition function for the state variable ${(e^c(h))}_{c\in C}$ is derived from the constraint (B.3). The amount of electricity in the ESS at the stage t + 1 depends on that at the stage t and the ESS charging and discharging amounts during the stage t. In addition, the forecast errors in amounts of solar PV generation and electricity demand in the state space evolve according to the exogenously determined respective uncertainty scenarios.

Immediate Reward Function

• • $f_1({\mathbf{s}}_1,{\mathbf{a}}_1)={\sum }_{t\in \{2,3,4,5\}}{\sum }_{h\in H_t}{\lambda }_{\mathit{SMP}}(h)q(h).$

• •   $f_t({\mathbf{s}}_t,{\mathbf{a}}_t)$

  $$\begin{array}{c}=\underset{A_1}{\underbrace{{\displaystyle \sum _{h\in H_t}{\lambda }_{\mathit{SMP}}}(h)\{q(h)-{\displaystyle \sum _{j\in J}\hspace{0.17em}}[Z^j(h)+△Z^j(h)]-{\displaystyle \sum _{c\in C}{v}^c}{(h)}^{+}\}}}\\ \underset{A_2}{\underbrace{-{\displaystyle \sum _{h\in H_t}{\lambda }_{\mathit{inc}}}(h)q(h)(1-I^{+}(h)-I^{-}(h))}}\\ \underset{A_3}{\underbrace{+\beta {\displaystyle \sum _{h\in E_t}{\displaystyle \sum _{c\in C}{\lambda }_{\mathit{elec}}}}(h)[G^c(h)+△G^c(h)-b^{c,dc}(h)+b^{c,ch}(h)]}}\hspace{0.17em}\hspace{0.17em}\forall t\in T^{\prime }.\end{array}$$

In the first stage, the immediate reward $f_1(s_1,a_1)$ represents the VPP’s revenue from DER sales as the bidding amounts. In the other stages, the immediate reward $f_t(s_t,a_t)$ denotes the costs during the stage $t\in T^{\prime }$. The term A₁ adjusts the revenue calculated at stage t = 1. The adjustment depends on the difference between the bidding and actual supply amounts. The actual supply is defined as the sum of the actual solar PV generation and DR amount in each hour. If the actual supply amount is insufficient (oversupplied) for the bidding amount at a certain hour, the term A₁ has a positive (negative) value. The term A₂ defines the balancing incentive in accordance with the bidding amounts. Last, the term A₃ determines the payments for electricity usage of companies in the VPP platform.

Objective Function

Our objective is to maximize the total expected profits of the VPP over all feasible decisions:

$$ma{x}_{\pi }{\displaystyle \sum _{t\in T}\mathbb{E}}[f_t({\mathbf{s}}_t,{\mathbf{a}}_t)]\hspace{0.17em},$$(B.12)

where π is a feasible policy that is a sequence of decisions. The optimal value of being state s_t in each stage t can be expressed by the Bellman optimality equations as follows:

$$\begin{array}{l}{V}_1({\mathbf{s}}_1)=ma{x}_{a_1}\{f_1({\mathbf{s}}_1,{\mathbf{a}}_1)-\mathbb{E}[V_2({\mathbf{s}}_2)|{\mathbf{s}}_1]\},\hspace{0.17em}\\ V_t({\mathbf{s}}_t)=mi{n}_{{\mathbf{a}}_t}\{f_t({\mathbf{s}}_t,{\mathbf{a}}_t)+\mathbb{E}[V_{t+1}({\mathbf{s}}_{t+1})|{\mathbf{s}}_t])\}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall t\in T^{\prime },\hspace{0.17em}\end{array}$$(B.13)

where we assume that $V_6(s_6)=0$.

Appendix C. Framework for Uncertainty Description of Prosumer’s Electricity Demand

H Energy Co. Ltd. develops a clustering-based model for the electricity demand forecast. The model uses the k-means clustering method with the Euclidean distance measure. The detailed procedure to forecast the electricity demand and to generate its scenario lattice based on the model is summarized as follows:

• Step 1. Prepare the historical data for the prosumer’s electricity demand.

• Step 2. Split the historical data into the data for each three-hour time period (stage).

• Step 3. For each time period, cluster the split data by using the k-means clustering method. The best number of clusters can vary among the time periods. We set k as 15.

• Step 4. Compute the centroid of each cluster.

• Step 5. Among the clusters, select the clusters that include the data for the recent three days.

• Step 6. Use the average of centroids of selected clusters as the forecast.

• Step 7. Generate the scenario lattice based on the selected clusters. The centroid of each selected cluster is used as a scenario value, and the transition probabilities between clusters in two successive time periods can be calculated based on historical trajectories for the recent three days. For example, in Figure 7, one scenario at stage 4 in the lattice implies that all data for the recent three days are included in the same cluster at stage 4. However, at stage 5, two days are categorized into the same cluster, but the remaining day is clustered into another cluster. As a result, the transition probabilities are calculated as two-thirds and one-third.

Appendix D Estimated Result of Solar PV Generation Forecasting Model

In our pilot test, before describing the uncertainty in the solar PV generation forecast, we develop its forecasting models as introduced in the Uncertainty Description section. The SARIMA model can be expressed by the combination of order $(p,d,q)\times {(P,D,Q)}_s$ in the model, where p is the nonseasonal autoregressive (AR) order, d is the nonseasonal differencing, q is the nonseasonal moving average (MA) order, P is the seasonal AR order, D is the seasonal differencing, Q is the seasonal MA order, and s is the time span of repeating seasonal pattern (Box et al. 2015). Because the solar PV generation has strong daily patterns, we set s as 24. We set the ranges of candidate orders for the other parameters between one and three for simplicity and find the best combination of orders based on the Akaike information criterion (AIC) score. The estimated forms of forecasting models are shown in Table D.1. We note that four forecasting models are developed for two seasons and two bidding times (10 a.m. and 5 p.m.).

Table D.1. Orders of Seasonal ARIMA Models for Solar PV Generation Forecast Are Determined

Table D.1. Orders of Seasonal ARIMA Models for Solar PV Generation Forecast Are Determined

$(p,d,q)\times {(P,D,Q)}_s$	For 10 a.m.	For 5 p.m.
Summer	$(1,1,2)\times {(3,1,3)}_{24}$	$(1,1,2)\times {(3,1,3)}_{24}$
Winter	$(2,1,3)\times {(3,1,3)}_{24}$	$(3,1,3)\times {(3,1,1)}_{24}$

Appendix E. Tightening Big-M Heuristic Method

For tightening $M_1(h)$, we logically infer a greatest value for the left-hand side of each inequality (B.5) and (B.6). Because ${[\overline{d^c(h)}-d_{\omega }^c(h)]}^{+}$ has nonnative value, the tightest inequality from Equations (B.5) and (B.6) can be written as

$$q(h)-{\displaystyle \sum _{j\in J}\hspace{0.17em}}[Z^j(h)+△Z_{\omega }^j(h)]\le \alpha \rho (1+M_1(h)*I^{+}(h))\hspace{0.17em}\forall h\in H_t\hspace{0.17em},\forall t\in T^{\prime },$$(E.1)

$${\displaystyle \sum _{j\in J}\hspace{0.17em}}[Z^j(h)+△Z_{\omega }^j(h)]+{\displaystyle \sum _{c\in C}\hspace{0.17em}}{[\overline{d^c(h)}-G^c(h)-△G_{\omega }^c(h)+B_{\mathit{max}}^c]}^{+}-q(h)\le \alpha \rho (1+M_1(h)*I^{-}(h))\hspace{0.17em}\forall h\in H_t\hspace{0.17em},\forall t\in T^{\prime }.$$(E.2)

Let us define $A_{\mathit{tight}}(h),\hspace{0.17em}{B}_{\mathit{tight}}(h)$ as

$$A_{\mathit{tight}}(h)=\frac{q(h)-\sum _{j\in J}\hspace{0.17em}[Z^j(h)+△Z_{\omega }^j(h)]-\alpha \rho }{\alpha \rho }\forall h\in H_t\hspace{0.17em},\forall t\in T^{\prime },$$(E.3)

$$B_{\mathit{tight}}(h)=\frac{\sum _{j\in J}\hspace{0.17em}[Z^j(h)+△Z_{\omega }^j(h)]+\sum _{c\in C}\hspace{0.17em}{[\overline{d^c(h)}-G^c(h)-△G_{\omega }^c(h)+B_{\mathit{max}}^c]}^{+}-q(h)-\alpha \rho }{\alpha \rho }\forall h\in H_t\hspace{0.17em},\forall t\in T^{\prime }.$$(E.4)

We apply the tightened big-M values for $M_1(h)$ by defining $M_1(h)=\mathit{max}\{A_{\mathit{tight}}(h),B_{\mathit{tight}}(h)\}$.

In a similar way, we can tighten the big-M value for M₂ as

$$C_{\mathit{tight}}(h)=\left|\overline{d^c(h)}-\{G^c(h)+△G_{\omega }^c(h)+B_{\mathit{max}}^c\}\right|\forall \mathit{\text{c}}\in \mathit{\text{C}}\hspace{0.17em}\forall \mathit{\text{h}}\in {\mathit{\text{H}}}_{\text{t}}\hspace{0.17em},\forall \mathit{\text{t}}\in {\mathit{\text{T}}}^{\prime },$$(E.5)

$$D_{\mathit{tight}}(h)=\left|\overline{d^c(h)}-\{G^c(h)+△G_{\omega }^c(h)-B_{\mathit{max}}^c\}\right|\forall \mathit{\text{c}}\in \mathit{\text{C}}\hspace{0.17em}\forall \mathit{\text{h}}\in {\mathit{\text{H}}}_{\text{t}}\hspace{0.17em},\forall \mathit{\text{t}}\in {\mathit{\text{T}}}^{\prime }.$$(E.6)

We apply the tightened big-M values for $M_2(h)$ by defining $M_2(h)=\mathit{max}\{C_{\mathit{tight}}(h),D_{\mathit{tight}}(h)\}$.

Appendix F. Fixing A Priori Heuristic Method

Based on the CBL and usage data, we fix the binary variables in advance that are likely to be determined before the operation. During the test, we found that no charge was made during the peak load period and little discharge was found during the light load period. Therefore, we prefix the binary variable associated with DR as

$$\text{In}\hspace{0.17em}\text{peak}\hspace{0.17em}\text{load}\hspace{0.17em}\text{period},\hspace{0.17em}\hspace{0.17em}\text{if}\hspace{0.17em}\overline{d^c(h)}-G^c(h)+△G_{\omega }^c(h)>0\hspace{0.17em},\hspace{0.17em}v{i}_{\omega }^c{(h)}^{+}=1,$$(F.1)

$$\text{In}\hspace{0.17em}\text{light}\hspace{0.17em}\text{load}\hspace{0.17em}\text{period},\hspace{0.17em}\hspace{0.17em}\text{if}\hspace{0.17em}\overline{d^c(h)}-G^c(h)+△G_{\omega }^c(h)<0\hspace{0.17em},\hspace{0.17em}v{i}_{\omega }^c{(h)}^{\text{-}}=1.$$(F.2)

Verification Letter

Mr. Ilhan Ham, CEO and Founder, H Energy Company, Ltd, 64 Jigok-Ro, Nam-Gu, Pohang, Gyeongbuk 37666, Republic of Korea, writes:

“This letter is to confirm the implementation of the optimization model and data analytics approach at H Energy Co., Ltd. described in the article, “Data Analytics and Optimization for the Optimal Bidding of a Virtual Power Plant.” In an era when the production and transaction of renewable energy are being activated, H Energy Co., Ltd tries to create the values by collecting small distributed energy resources (DERs) and intermediating between DERs and the electricity market based on a cloud-based IT platform. The model and solution approach proposed through this project are implemented in our platform, and those provide the quantitative and qualitative benefits about the operation of our business model. In addition, this project helps us to understand the structure of DER market and how to react in the process of revising the market structures. Ultimately, in the long term, this project will allows us to play a leading role in the market and will also contribute to the overall welfare of society by improving the efficiency of the energy market.”