A Robust Optimization Model of Aggregated Resources Considering Serving Ratio for Providing Reserve Power in the Joint Electricity Market

Abstract

As the share of distributed generation increases, so do the opportunities for aggregators to participate in the electricity market. In particular, aggregators participating in both the day-ahead and real-time markets contribute to improving the reliability of the power system. In addition, aggregators seeking additional revenue can benefit from providing reserves in a joint electricity market environment. However, aggregated resources with uncertainty are limited because of the uncertain nature of both reserve provision and the amount of reserves they can provide. Therefore, this study proposes a robust optimization model for an aggregator to formulate a strategy for participation in the day-ahead markets and deploys energy control in the real-time operation. The serving ratio reflects the availability of the aggregator’s reserve participation. Both the deployed up/down power and renewable energy in the real-time operation are considered as uncertain parameters to reflect the uncertainty. In the case study, we analyze the profit-maximization strategy of an aggregator that owns renewable energy resources and energy-storage systems under the variation interval for uncertain parameters and the serving ratio. The bidding strategies vary by the variation interval and the serving ratio.

1. Introduction

Due to the reduction in carbon dioxide emissions through the decrease in conventional fossil fuels and the decentralized operation of power grids, distributed generation of electricity is being actively adopted. The adoption of renewable energy and gas power generation is not only helping to balance supply and demand across the power system, but also positively impacting ramping capabilities in response to system volatility [1]. Distributed generation also has the advantage of small capacity and easy installation near demand locations; thus, they are often installed in multiple locations. In addition, as the cost of distributed generation has decreased due to technological advancements, operators or aggregators have emerged to own and manage distributed generation. Implementing demand-friendly policies can significantly increase the capacity of resources near the demand, thereby reducing resource costs and improving energy efficiency [2,3,4]. This is illustrated in Figure 1. As aggregators’ capacity becomes comparable to that of conventional generators, the former can participate in the operation of both electricity markets and the power grid. Furthermore, with the emergence of rules allowing distributed generation and energy storage to participate in the electricity market, aggregators’ participation in the electricity market is expected to increase.

Aggregators essentially participate in the day-ahead market for electricity, bidding on a forecast of how much they will provide to the grid. Unlike traditional generators’ forecasts, forecasts for distributed generation are subject to volatility and uncertainty. Previously, when the share of distributed generation was small, the electricity market could operate regardless of the accuracy of this forecast. However, in the present or future, when the share of distributed generation is large, the accuracy of the forecast may adversely affect the operation of the electricity market. To mitigate these effects, aggregators may utilize methods to reduce uncertainty by increasing the number of distributed generation to smooth out output fluctuations or by increasing forecast accuracy [5,6]. Because the unit cost of distributed generation is lower than that of conventional generators, aggregators mostly participate in the day-ahead market as price takers. Aggregators want to plan the scheduling of their resources in response to the given or expected day-ahead market prices to maximize their profits [7,8,9,10,11]. In scheduling planning, uncontrollable renewable energy resources determine the planned amount through forecasting, while controllable distributed generation and energy-storage systems determine the planned amount by adjusting their output or establishing charging and discharging plans. Through the sum of the uncontrollable and controllable scheduled amounts, the aggregators’ optimal bidding strategy is proposed [7,8,10]. Aggregators are able to have a strategy and adjust their output on a day-ahead stage.

As the share of conventional generators declines, aggregators participate in not only the day-ahead market but also the regulation or reserve markets [12]. As aggregated resources are required to contribute to the stable operation of power systems such as conventional generators, a joint market-optimization model is proposed in which aggregators consider both the energy and reserve markets in their scheduling plans. The energy market clears or settles for the amount of generation, and the reserve market clears or settles for the opportunity cost of insufficient or excess generation. Basically, the optimization problem for the energy and reserve markets is constructed from the perspective of a grid operator using reserve power [13]. However, depending on the grid and market-operating regime, aggregators may be required to build and purchase reserves to ensure the reliable operation of their generation units [14]. In this situation, aggregators do not trade their power only on the energy market. It is more profitable for them to participate in the reserve market, even if they do not generate power in the power plants they manage. Therefore, many studies have proposed a solution to the joint-market optimization problem in which the aggregators’ profit is maximized [14,15,16,17,18] and the distribution system’s operation cost is minimized [19,20,21]. In the studies on maximizing aggregators’ profits, bidding scheduling in the energy and reserve markets was optimized by considering the use of energy storage, demand response, etc., which aggregators could control and the uncertainty of renewable energy. In the studies on minimizing distribution-system operating costs, measures are proposed to resolve the uncertainty of the system while securing and utilizing reserve power that can match supply with the demand. Therefore, aggregators’ participation in the reserve market can help increase their profits while improving the reliability of the power system.

When reserve power is used in a real-time operation, it is settled at the real-time market price; that is, the amount of reserve power used is settled as energy in the real-time operation. The amount of reserve power is an important factor in optimizing the income or cost for aggregators who receive the difference between the day-ahead market and real-time operation. In addition, in an environment in which the value of upward or downward reserve power is evaluated differently depending on the type of reserve power, it is necessary to establish an optimal strategy for securing and utilizing the two opposite reserve powers [22]. The inherent nature of aggregators with uncertain distributed generation leads to uncertainty about the amount of output they provide in the real-time operation. This uncertainty, together with uncertainty in the value of reserves, can exacerbate the discrepancy between the day-ahead market and real-time operation. To mitigate this uncertainty, studies have been proposed that consider the uncertainty of renewable-energy output and real-time market prices [7,23,24]. Aggregators exposed to uncertainty are likely to suffer losses due to penalties when settling in the real-time operation. Therefore, an optimization problem that considers settlement in the real-time operation may be able to maximize profits while minimizing losses for aggregators.

One way to consider uncertainty is the robust optimization method, which is generally used to derive optimal results by considering the worst-case scenario for optimization variables with uncertainty. Based on the characteristics of the robust optimization problem, the robust optimization used in the study establishes an optimal operation plan by considering the risk of variables with uncertainty, such as power generation and price in the real-time operation [8,9,10,11,24,25,26,27,28]. In a situation in which an aggregator who uses wind turbine and energy storage together participates in the energy and reserve markets, an optimization model that considers the uncertainties of both wind-power generation and reserve-power provision is introduced [25]. Accordingly, the robust optimization model used in this study considers the uncertainties of wind-power generation in the real-time operation and reserve-power provision using energy storage. Meanwhile, the robust optimization model that considers price uncertainty in the real-time operation also analyzes the trade-off characteristics of maximizing aggregators’ profit and the real-time operation feasibility of the power system [26]. By considering the driving pattern of electric vehicle owners as an uncertainty, a robust optimization method was proposed to secure reserve power used in the day-ahead market and real-time operation through charging and discharging of electric vehicle chargers [27]. On the other hand, a robust optimization method was proposed, taking into account uncertainties such as power system failures, renewable energy fluctuations, and demand forecast errors. This method also considers participation in both the energy market and the reserve power market, as mentioned in [28]. Recent studies have shown a trend toward including information on securing both up and down reserve capacity, in addition to the reserve capacity provided in the upward direction. Because robust optimization methods that consider risk can lead to better aggregators’ profit, it seems that robust optimization methods can be appropriately used in aggregators’ optimization models that consider the real-time operation.

There are various methods for solving optimization problems, but mixed-integer linear programming (MILP), which employs linearized objective functions and constraints, is frequently preferred for utilizing commercial software and reducing calculation time. While robust optimization with complex uncertainty constraints may be addressed using nonlinear objective functions and constraints [28], it can also be solved by linearizing the objective function and constraints and employing MILP techniques [25,27]. Additionally, linearizing the objective function and constraints helps find a mathematically guaranteed global optimum.

Previous studies have shown that aggregators with uncertainty in their outputs develop optimal scheduling plans for energy and reserves under a joint market, considering the settlement of real-time price and robust optimization, as shown in

Table 1. Overview of the surveyed literature on optimization models in this study.

Research	Energy	Reserve	Settlement of Real-Time Price	Robust Model	Serving Ratio for Reserve	Markets *
[7]	O	X	X	X	X	DA
[8,9,10]	O	X	X	O	X	DA
[11]	O	X	O	O	X	DA
[12,13,14,15,16,17,18,19,20,21]	O	O	X	X	X	DA and RM
[22,23]	O	O	O	X	X	DA, RM, and RT
[24]	O	O	O	O	X	DA, RM, and RT
[25,27,28]	O	O	X	O	X	DA and RM
[26]	O	O	O	O	X	DA, RM, and RT
This study	O	O	O	O	O	DA, RM, and RT

* DA: day-head market, RM: reserve market, RT: real-time operation.

. However, due to the nature of aggregators with uncertainty, there are limitations in providing reserve power. Although distributed generation can provide reserves [12], they can only do so to a limited extent. Thus, it should be possible to consider the proportion of reserve provision in the available capacity of the total aggregated resources when planning aggregators’ energy and reserve scheduling. To the best of the authors’ knowledge, no previous research has considered the proportion of reserve provision. Therefore, in this study, we propose a robust optimization model that considers settlement in real-time operation in the joint market according to the proportion of reserve provision desired by the aggregator. Based on the robust optimization model proposed in [25], we consider the difference between the day-ahead market and real-time operation, along with an uncertain parameter that reflects the share of reserve provision in planned generation rather than that based on historical data. Our robust optimization model is formulated as the MILP. The main contributions of this study can be summarized as follows:

• We propose a bidding strategy for renewable energy resources and energy storage owned by an aggregator who can participate in both the energy and reserve markets;
• The aggregator can provide reserve power as a percentage of the generation determined in the day-ahead market, and the reserve power can be utilized to increase or decrease generation in the real-time operation;
• Unlike previous studies that only used real-time reserve prices for reserve settlement, this study uses the real-time market price for reserve settlement to pay for increased or decreased generation;
• We introduce an optimization problem based on robust optimization, which considers that uncertainties of renewable energy and power-generation increase and decrease in the real-time operation.

The remainder of the paper is organized as follows: In Section 2, we introduce the joint-market participation of aggregated resources. In Section 3, we introduce a robust optimization model that reflects the aggregator’s bidding strategy. In Section 4, we present the simulation results. Finally, in Section 5 and Section 6, we discuss our findings and conclude the paper, respectively.

2. Aggregated Resources’ Participation in the Joint Market

2.1. Assumptions

This study proposes an optimization model to plan the energy and reserve together in the day-ahead and real-time markets according to the serving ratio for reserve power specified in advance by the aggregator. The model assumptions are as follows:

• The day-ahead market structure comprises an energy market and a reserve market as proposed in [25]. The reserve market is a market for trading reserve power to be used on the delivery day. Reserve power is divided into up- and down-reserve power: in the real-time operation, up-reserve power is used to generate additional energy from the planned energy, and down-reserve power is used to decrease the planned energy.
• The aggregator determines the available capacity based on the forecast output in the day-ahead operation and how much of the available capacity can be used to provide reserve power by assessing the serving ratio. The serving ratio applies only to the day-ahead operation and, once determined, does not change in the real-time operation.
• As with [25], by considering the real-time operation, the aggregator can expect to increase or decrease their output within the determined reserve power in the day-ahead market. The system operator can accept any increase or decrease in output from the aggregator.
• The real-time market uses a time step smaller than one hour, as opposed to the day-ahead market, which uses an hourly time step.
• All increases and decreases in output are settled at the real-time market price. However, a decrease in output except for a power imbalance in renewable energy resources between the day-ahead market and real-time operation is settled at the reserve price in the real-time market.

With respect to the above assumptions, we can show the market structure for the aggregators’ bidding strategy in Figure 2 and provide an overview of the optimization model configuration in Figure 3. In Figure 2, the market structure offers an overview of how the aggregator participates in both the day-ahead market and real-time operations, providing energy and reserves. The energy and reserves planned in the day-ahead market could be utilized as deployed power and deployed up/down power in real-time operations. In Figure 3, the configuration for available capacity and serving ratio in the optimization model conveys information about the decision variables examined in this study. When the serving ratio is determined with a given aggregator capacity, the optimization problem results in the determination of day-ahead market energy and reserves. It also involves determining real-time operations, taking uncertainty into account based on the previously determined day-ahead reserves, which, in turn, determine deployed power.

2.2. Participation Model Description

Based on the assumptions above, the aggregator’s electricity-market participation model is written as shown in Figure 4.

As shown in this figure, the serving ratio determines the capacity range for day-ahead energy and reserve power used in the real-time operations. A serving ratio of 0 means that the aggregator can only participate in the energy bidding in the day-ahead market, while a serving ratio of 1 indicates that the whole range of aggregator’s available capacity can be used as the reserve power for deployment in the real-time operations. Even when the serving ratio is 1, the aggregator does not use its whole capacity as reserve power. Instead, it can freely allocate energy and reserve power within its total output capacity based on its bidding strategy. The planned energy and reserve are utilized and settled in the day-ahead and real-time markets. First, in the day-ahead market, hourly energy and reserve planning and clearing or settlement are performed. In the real-time market, up-reserve power is used when the aggregator increases output, and down-reserve power is used when the aggregator decreases output against the reserve determined in the day-ahead market. This utilized reserve power creates a difference between the amount of energy utilized in the day-ahead market and that in the real-time operation, and the difference is settled through the real-time market price.

A detailed description of the robust optimization model is as follows:

• The aggregator schedules their output to maximize profits by considering the day-ahead market and real-time operation. The aggregator predicts the amount of renewable energy utilized in the real-time operation and the amount of energy and reserves utilized in the electricity market by charging and discharging energy-storage systems;
• The reserve power planned by the day-ahead market is determined by the sum of the hourly forecast power of renewable energy resources and maximum power of energy storage multiplied by the serving ratio for reserve power;
• Uncertainties that cause the difference between the day-ahead market and real-time operation comprise the deployed up/down power and renewable energy in the real-time operation.

3. Robust Optimization Model for Aggregated Resources Considering the Serving Ratio

The following equations in this section are derived from [25]. While some symbols, variables, and equations may bear similarities to those in [25], they have been adjusted to align with the focus of this study, incorporating serving ratio and robust optimization components.

3.1. Objective Function

The objective function of the deterministic model has the operating-income and operating-cost functions for maximizing the aggregator’s profit, as shown in Equation (1):

$$\begin{array}{c}\hfill Max\sum _{\forall t}\left(I_t-C_t\right),\end{array}$$

(1)

where $I_t$ and $C_t$ are the aggregator’s operating-income and operating-cost functions at time t, respectively.

The operating-income function comprises the day-ahead and real-time income from selling and buying power or reserve in the energy and reserve markets. In the day-ahead market, the aggregator can obtain income by selling the generated power and bidding the reserve. Meanwhile, the aggregator’s income is reduced by purchasing the generated power from the day-ahead market for charging the energy storage. Within the real-time operation, power imbalances between the day-ahead and real-time markets can arise due to the deployed power in the up- or down-reserve services and uncertain power from renewable energy resources. The deployed power in the up-reserve services can receive the real-time price. On the other hand, the deployed power in the down-reserve services may incur losses, as it does not receive the real-time price; however, it can mitigate some of the losses by receiving the real-time reserve price.

From the above explanation, the aggregator’s operating-income function is formulated by Equation (2):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_t=& \left({\pi }_t^{DA}·P_t^{DA,S}-{\pi }_t^{DA}·P_t^{DA,B}+{\pi }_t^{DARS}·P_t^{RS}\right)\hfill \\ \hfill & +\sum _{j=1}^{N_j}\Delta S_{N_j}·\left({\pi }_{t,j}^{RT}·\left(P_{t,j}^{Up}-P_{t,j}^{Down}-P_{t,j,r}^{RES,imb}\right)+{\pi }_{t,j}^{RTRS}·P_{t,j}^{Down}\right),\hfill \end{array}\end{array}$$

(2)

where the first and second lines represent the day-ahead and real-time income, respectively. The first line, representing the day-ahead income, comprises the net value of trading power in the energy market and the awarded reserve in the reserve market. The second line, representing the real-time income, is calculated based on imbalances in the real-time market and compensation of the down-reserve service in the real-time reserve market. In the real-time market, where calculations occur in minutes instead of hourly as in the day-ahead market, a duration is required to convert the energy quantity into hourly units. $\Delta S_{N_j}$ is the duration for the real-time market. This duration can be calculated by dividing the interval for the real-time market represented in minutes by 60 min.

The aggregator’s operating-cost function is formulated by Equation (3):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_t=& \sum _{s=1}^{N_s}\left({\rho }_s^{DCH}·P_{t,N_j,s}^{DA,DCH}+{\rho }_s^{CH}·P_{t,N_j,s}^{DA,CH}\right)+\sum _{r=1}^{N_r}{\rho }_r·P_{t,N_j,r}^{DA,Energy}\hfill \\ \hfill & +\sum _{j=1}^{N_j}\Delta S_{N_j}·(\sum _{s=1}^{N_s}\left({\rho }_s^{DCH}·(P_{t,j,s}^{Up,DCH}+P_{t,j,s}^{Down,DCH})+{\rho }_s^{CH}·(P_{t,j,s}^{Up,CH}+P_{t,j,s}^{Down,CH})\right)\hfill \\ \hfill & +\sum _{r=1}^{N_r}{\rho }_r·(P_{t,j,r}^{Up,RES}-P_{t,j,r}^{Down,RES}));\forall t\in T,\hfill \end{array}\end{array}$$

(3)

where the first, second, and third lines represent the costs of providing energy in the day-ahead market, charging and discharging energy storage in the real-time operation, and generating renewable energy in the real-time operation, respectively. The left term in the first line represents the charging and discharging cost of energy storage in the day-ahead market. The right term in the first line represents the generation cost of the renewable energy in the day-ahead market. $N_j$ in the first term is utilized to represent the fixed traded energy in the intra-hour periods. In the second and third lines, the deployed power in minutes is converted into the hourly deployed power by $\Delta S_{N_j}$. Costs of the deployed discharging/charging power in the real-time up and down operation, as shown in the second line, are settled by ${\rho }_s^{DCH}$ and ${\rho }_s^{CH}$, respectively. The aggregator is assumed to be able to discharge/charge power in the real-time down/up operation ($P_{t,j,s}^{Down,DCH}$/$P_{t,j,s}^{Up,CH}$). While real-time down/up operations require power reductions/increases, aggregators can strategically control discharging/charging power to adjust income or costs. This representation is different from that of the cost described in [25]. The power generation of renewable energy resources in the real-time up and down operation is settled by ${\rho }_r$, as shown in the third line.

• Constraints on day-ahead energy schedule:

The day-ahead energy sales and purchases comprise the discharging/charging power and renewable energy, and these constraints are fixed in the intra-hour periods, as shown in Equations (4)–(8):

$$\begin{array}{c}\hfill P_t^{DA,S}=\sum _{s=1}^{N_s}{P}_{t,N_j,s}^{DA,DCH}+\sum _{r=1}^{N_r}{P}_{t,N_j,r}^{DA,Energy};\forall t\in T,\end{array}$$

(4)

$$\begin{array}{c}\hfill P_t^{DA,B}=\sum _{s=1}^{N_s}{P}_{t,N_j,s}^{DA,CH};\forall t\in T,\end{array}$$

(5)

$$\begin{array}{c}\hfill P_{t,j,s}^{DA,DCH}=P_{t,j^{\prime },s}^{DA,DCH};\forall t\in T,\forall s\in S,\forall j,j^{\prime }\in J,\end{array}$$

(6)

$$\begin{array}{c}\hfill P_{t,j,s}^{DA,CH}=P_{t,j^{\prime },s}^{DA,CH};\forall t\in T,\forall s\in S,\forall j,j^{\prime }\in J,\end{array}$$

(7)

$$\begin{array}{c}\hfill P_{t,j,r}^{DA,Energy}=P_{t,j^{\prime },r}^{DA,Energy};\forall t\in T,\forall r\in R,\forall j,j^{\prime }\in J.\end{array}$$

(8)

• Constraints on day-ahead reserve schedule:

The day-ahead reserve comprises the potential power from charging/ discharging power and renewable energy, and these constraints are also fixed in the intra-hour periods, as shown in Equations (9)–(12):

$$\begin{array}{c}\hfill P_t^{RS}=\sum _{s=1}^{N_s}\left(P_{t,N_j,s}^{RS,CH}+P_{t,N_j,s}^{RS,DCH}\right)+\sum _{r=1}^{N_r}\left(P_{t,N_j,r}^{RS,RES}\right);\forall t\in T,\end{array}$$

(9)

$$\begin{array}{c}\hfill P_{t,j,s}^{RS,CH}=P_{t,j^{\prime },s}^{RS,CH};\forall t\in T,\forall j,j^{\prime }\in J,\forall s\in S,\end{array}$$

(10)

$$\begin{array}{c}\hfill P_{t,j,s}^{RS,DCH}=P_{t,j^{\prime },s}^{RS,DCH};\forall t\in T,\forall j,j^{\prime }\in J,\forall s\in S,\end{array}$$

(11)

$$\begin{array}{c}\hfill P_{t,j,r}^{RS,RES}=P_{t,j^{\prime },r}^{RS,RES};\forall t\in T,\forall j,j^{\prime }\in J,\forall r\in R.\end{array}$$

(12)

• Constraints on deployed up and down power in the real-time operation:

Based on the determined reserve power in the day-ahead market, the aggregator can control the deployed power in the real-time up and down operations. Therefore, the deployed up/down power in the real-time operation is capped by the determined reserve power in the day-ahead market, as shown in Equations (13) and (14):

$$\begin{array}{c}\hfill P_{t,j}^{Up}\le P_t^{RS};\forall t\in T,\forall j\in J,\end{array}$$

(13)

$$\begin{array}{c}\hfill P_{t,j}^{Down}\le P_t^{RS};\forall t\in T,\forall \in J.\end{array}$$

(14)

The deployed up power in the real-time operation comprises the discharging power of energy storage and increased power of the renewable energy resources. However, the charging power decreases the deployed up power. These constraints are described in Equation (15):

$$\begin{array}{c}\hfill P_{t,j}^{Up}=\sum _{s=1}^{N_s}\left(P_{t,j,s}^{Up,DCH}-P_{t,j,s}^{Up,CH}\right)+\sum _{r=1}^{N_r}{P}_{t,j,r}^{Up,RES};\forall t\in T,\forall j\in J.\end{array}$$

(15)

In contrast to the above deployed up power, the deployed down power in the real-time operation comprises the charging power of energy storage and decreased power of the renewable energy resources. The discharging power decreases the deployed down power. These constraints are represented in Equation (16):

$$\begin{array}{c}\hfill P_{t,j}^{Down}=\sum _{s=1}^{N_s}\left(P_{t,j,s}^{Down,CH}-P_{t,j,s}^{Down,DCH}\right)+\sum _{r=1}^{N_r}{P}_{t,j,r}^{Down,RES};\forall t\in T,\forall j\in J.\end{array}$$

(16)

• Constraints on power imbalance of renewable energy resources:

With the uncertain power of renewable energy resources, the imbalance in power between the day-ahead market and real-time operation would be considered. The power imbalance is calculated by subtracting the day-ahead energy scheduling and deployed power of renewable energy in the real-time up and down operation from actual power generation, as shown in Equations (17) and (18):

$$\begin{array}{c}\hfill P_{t,j,r}^{RES,imb}=P_{t,j,r}^{RT}-\left(P_{t,j,r}^{DA,Energy}+P_{t,j,r}^{Up,RES}-P_{t,j,r}^{Down,RES}\right);\forall t\in T,\forall j\in J,\forall r\in R,\end{array}$$

(17)

$$\begin{array}{c}\hfill 0\le P_{t,j,r}^{RES,imb}\le P_{t,j,r}^{RT};\forall t\in T,\forall j\in J,\forall r\in R.\end{array}$$

(18)

3.2. Operation Constraints

The operation constraints of energy storage are considered in the day-ahead market and real-time operation. These constraints comprise minimum/maximum charging/discharging power, ramp-rate characteristics, and deployed power in the real-time up and down operation considering energy and reserve, as shown in Equations (19)–(21) for the decision operation for charging and discharging modes, Equations (22)–(30) for the charging operation, Equations (31)–(39) for the discharging operation, and Equations (40) and (41) for stored energy.

• Constraints on energy storage operation:

$$\begin{array}{c}\hfill {\alpha }_{t,j,s}={\alpha }_{t,j^{\prime },s};\forall t\in T,\forall j,j^{\prime }\in J,\forall r\in R,\end{array}$$

(19)

$$\begin{array}{c}\hfill {\beta }_{t,j,s}={\beta }_{t,j^{\prime },s};\forall t\in T,\forall j,j^{\prime }\in J,\forall s\in S,\end{array}$$

(20)

$$\begin{array}{c}\hfill 0\le {\alpha }_{t,j,s}+{\beta }_{t,j,s}\le 1;\forall t\in T,\forall j\in J,\forall s\in S.\end{array}$$

(21)

• Operation constraints on charging of energy storage:

$$\begin{array}{c}\hfill P_s^{min}·{\alpha }_{t,j,s}\le P_{t,j,s}^{DA,CH}\le P_s^{max}·{\alpha }_{t,j,s};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(22)

$$\begin{array}{c}\hfill P_s^{min}\le P_{t,j,s}^{RS,CH}\le P_s^{max}·{\alpha }_{t,j,s}-P_{t,j,s}^{DA,CH};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(23)

$$\begin{array}{c}\hfill P_{t,j,s}^{DA,CH}+P_{t,j,s}^{RS,CH}\le P_s^{max}·{\alpha }_{t,j,s};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(24)

$$\begin{array}{c}\hfill P_s^{min}·{\alpha }_{t,j,s}\le P_{t,j,s}^{DA,CH}-P_{t,j,s}^{RS,CH};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(25)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-R_s\le P_{t,j,s}^{DA,CH}\le R_s,\hfill & \mathrm{if} t=1,j=1,\hfill \\ -R_s\le P_{t,j,s}^{DA,CH}-P_{t-1,N_j,s}^{DA,CH}\le R_s,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ -R_s\le P_{t,j,s}^{DA,CH}-P_{t,j-1,s}^{DA,CH}\le R_s,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall s\in S,\end{array}$$

(26)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{P}_{t,j,s}^{RS,CH}\le R_s,\hfill & \mathrm{if} t=1,j=1,\hfill \\ P_{t,j,s}^{RS,CH}+P_{t-1,N_j,s}^{RS,CH}\le R_s,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ P_{t,j,s}^{RS,CH}+P_{t,j-1,s}^{RS,CH}\le R_s,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall s\in S,\end{array}$$

(27)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}-R_s\le P_{t,j,s}^{DA,CH}+P_{t,j,s}^{RS,CH}\le R_s,\hfill & \mathrm{if} t=1,j=1,\hfill \\ -R_s\le \left(P_{t,j,s}^{DA,CH}-P_{t-1,N_j,s}^{DA,CH}\right)+\left(P_{t,j,s}^{RS,CH}+P_{t-1,N_j,s}^{RS,CH}\right)\le R_s,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ -R_s\le \left(P_{t,j,s}^{DA,CH}-P_{t,j-1,s}^{DA,CH}\right)+\left(P_{t,j,s}^{RS,CH}+P_{t,j-1,s}^{RS,CH}\right)\le R_s,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall s\in S,\end{array}\end{array}$$

(28)

$$\begin{array}{c}\hfill P_{t,j,s}^{Up,CH}\le P_{t,j,s}^{RS,CH};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(29)

$$\begin{array}{c}\hfill P_{t,j,s}^{Down,CH}\le P_{t,j,s}^{RS,CH};\forall t\in T,\forall j\in J,\forall s\in S.\end{array}$$

(30)

• Operation constraints on discharging of energy storage:

$$\begin{array}{c}\hfill P_s^{min}·{\beta }_{t,j,s}\le P_{t,j,s}^{DA,DCH}\le P_s^{max}·{\beta }_{t,j,s};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(31)

$$\begin{array}{c}\hfill P_s^{min}\le P_{t,j,s}^{RS,DCH}\le P_s^{max}·{\beta }_{t,j,s}-P_{t,j,s}^{DA,DCH};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(32)

$$\begin{array}{c}\hfill P_{t,j,s}^{DA,DCH}+P_{t,j,s}^{RS,DCH}\le P_s^{max}·{\beta }_{t,j,s};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(33)

$$\begin{array}{c}\hfill P_s^{min}·{\beta }_{t,j,s}\le P_{t,j,s}^{DA,DCH}-P_{t,j,s}^{RS,DCH};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(34)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-R_s\le P_{t,j,s}^{DA,DCH}\le R_s,\hfill & \mathrm{if} t=1,j=1,\hfill \\ -R_s\le P_{t,j,s}^{DA,DCH}-P_{t-1,N_j,s}^{DA,DCH}\le R_s,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ -R_s\le P_{t,j,s}^{DA,DCH}-P_{t,j-1,s}^{DA,DCH}\le R_s,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall s\in S,\end{array}$$

(35)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{P}_{t,j,s}^{RS,DCH}\le R_s,\hfill & \mathrm{if} t=1,j=1,\hfill \\ P_{t,j,s}^{RS,DCH}+P_{t-1,N_j,s}^{RS,DCH}\le R_s,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ P_{t,j,s}^{RS,DCH}+P_{t,j-1,s}^{RS,DCH}\le R_s,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall s\in S,\end{array}$$

(36)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}-R_s\le P_{t,j,s}^{DA,DCH}+P_{t,j,s}^{RS,DCH}\le R_s,\hfill & \mathrm{if} t=1,j=1,\hfill \\ -R_s\le \left(P_{t,j,s}^{DA,DCH}-P_{t-1,N_j,s}^{DA,DCH}\right)+\left(P_{t,j,s}^{RS,DCH}+P_{t-1,N_j,s}^{RS,DCH}\right)\le R_s,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ -R_s\le \left(P_{t,j,s}^{DA,DCH}-P_{t,j-1,s}^{DA,DCH}\right)+\left(P_{t,j,s}^{RS,DCH}+P_{t,j-1,s}^{RS,DCH}\right)\le R_s,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall s\in S,\end{array}\end{array}$$

(37)

$$\begin{array}{c}\hfill P_{t,j,s}^{Up,DCH}\le P_{t,j,s}^{RS,DCH};\forall t\in T,\forall j\in J,\forall s\in S,\end{array}$$

(38)

$$\begin{array}{c}\hfill P_{t,j,s}^{Down,DCH}\le P_{t,j,s}^{RS,DCH};\forall t\in T,\forall j\in J,\forall s\in S.\end{array}$$

(39)

• Operation constraints on stored energy of energy storage:

$$\begin{array}{c}\hfill E_{t,j,s}=\left\{\begin{array}{cc}{E}_{t,j-1,s}+\Delta S_{N_j}·\left(\begin{array}{cc}\hfill & P_{t,j,s}^{DA,CH}-P_{t,j,s}^{DA,DCH}\hfill \\ \hfill +& P_{t,j,s}^{Up,CH}+P_{t,j,s}^{Down,CH}\hfill \\ \hfill -& P_{t,j,s}^{Up,DCH}-P_{t,j,s}^{Down,DCH}\hfill \end{array}\right),\hfill & \mathrm{if} t\ge 1, j\ge 2,\hfill \\ E_{t-1,N_j,s}+\Delta S_{N_j}·\left(\begin{array}{cc}\hfill & P_{t,j,s}^{DA,CH}-P_{t,j,s}^{DA,DCH}\hfill \\ \hfill +& P_{t,j,s}^{Up,CH}+P_{t,j,s}^{Down,CH}\hfill \\ \hfill -& P_{t,j,s}^{Up,DCH}-P_{t,j,s}^{Down,DCH}\hfill \end{array}\right),\hfill & \mathrm{if} t\ge 2, j=1,\hfill \\ E_s^{max}/2,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

(40)

$$\begin{array}{c}\hfill E_s^{min}\le E_{t,j,s}\le E_s^{max};\forall t\in T,\forall j\in J,\forall s\in S.\end{array}$$

(41)

• Operation constraints on renewable energy resources:

The aggregator considers the operation constraints of the renewable energy in the day-ahead market and real-time operation. These constraints comprise forecast power, ramp-rate characteristics, and deployed power in the real-time up and down operation considering energy and reserve, as shown in Equations (42)–(50)

$$\begin{array}{c}\hfill 0\le P_{t,j,r}^{DA,Energy}\le P_{t,j,r}^{RT};\forall t\in T,\forall j\in J,\forall r\in R,\end{array}$$

(42)

$$\begin{array}{c}\hfill 0\le P_{t,j,r}^{RS,RES}\le P_{t,j,r}^{RT}-P_{t,j,r}^{DA,Energy};\forall t\in T,\forall j\in J,\forall r\in R,\end{array}$$

(43)

$$\begin{array}{c}\hfill P_{t,j,r}^{DA,Energy}+P_{t,j,r}^{RS,RES}\le P_{t,j,r}^{RT};\forall t\in T,\forall j\in J,\forall r\in R,\end{array}$$

(44)

$$\begin{array}{c}\hfill 0\le P_{t,j,r}^{DA,Energy}-P_{t,j,r}^{RS,RES};\forall t\in T,\forall j\in J,\forall r\in R,\end{array}$$

(45)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-R_r\le P_{t,j,r}^{DA,Energy}\le R_r,\hfill & \mathrm{if} t=1,j=1,\hfill \\ -R_r\le P_{t,j,r}^{DA,Energy}-P_{t-1,N_j,r}^{DA,Energy}\le R_r,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ -R_r\le P_{t,j,r}^{DA,Energy}-P_{t,j-1,r}^{DA,Energy}\le R_r,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall r\in R,\end{array}$$

(46)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{P}_{t,j,r}^{RS,RES}\le R_r,\hfill & \mathrm{if} t=1,j=1,\hfill \\ P_{t,j,r}^{RS,RES}+P_{t-1,N_j,r}^{RS,RES}\le R_r,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ P_{t,j,r}^{RS,RES}+P_{t,j-1,r}^{RS}\le R_r,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall r\in R,\end{array}$$

(47)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}-R_r\le P_{t,j,r}^{DA,Energy}+P_{t,j,r}^{RS,RES}\le R_r,\hfill & \mathrm{if} t=1,j=1,\hfill \\ -R_r\le \left(P_{t,j,r}^{DA,Energy}-P_{t-1,N_j,r}^{DA,Energy}\right)+\left(P_{t,j,r}^{RS,RES}+P_{t-1,N_j,r}^{RS,RES}\right)\le R_r,\hfill & \mathrm{if} t\ge 2,j=1,\hfill \\ -R_r\le \left(P_{t,j,r}^{DA,Energy}-P_{t,j-1,r}^{DA,Energy}\right)+\left(P_{t,j,r}^{RS,RES}+P_{t,j-1,r}^{RS,RES}\right)\le R_r,\hfill & \mathrm{otherwise},\hfill \end{array}\right.;\forall r\in R,\end{array}\end{array}$$

(48)

$$\begin{array}{c}\hfill 0\le P_{t,j,r}^{Up,RES}\le P_{t,j,r}^{RS,RES};\forall t\in T,\forall j\in J,\forall r\in R,\end{array}$$

(49)

$$\begin{array}{c}\hfill 0\le P_{t,j,r}^{Down,RES}\le P_{t,j,r}^{RS,RES};\forall t\in T,\forall j\in J,\forall r\in R.\end{array}$$

(50)

3.3. Serving Ratio for Reserve Power

The reserve power is capped by the serving ratio for reserve power. The aggregator can calculate the available capacity for providing reserve power using the hourly forecast power of renewable energy resources and maximum power of energy storage, as shown in Equations (51) and (52):

$$\begin{array}{c}\hfill P_t^{RS}\le \gamma ·P_t^{AvailCap};\forall t\in T,\end{array}$$

(51)

$$\begin{array}{c}\hfill P_t^{AvailCap}=\sum _{s=1}^{N_s}{P}_s^{max}+\sum _{r=1}^{N_r}{P}_{t,N_j,r}^{DA,Energy}.\end{array}$$

(52)

3.4. Robust Optimization Model

To consider the uncertain parameter in the real-time operation, the constraints of the uncertain parameter and transformed optimization problem should be represented. The uncertain parameters comprise the deployed up/down power from the reserve power and the actual power generation of renewable energy resources in the real-time operation. The uncertain parameters are assumed to take on values according to a symmetric distribution in the interval [$-\Delta R$, $\Delta R$]. The deployed up/down power in the real-time operations and actual power generation of renewable energy resources are represented in Equations (53)–(56), respectively.

• Constraints on uncertain parameters:

Deployed up power from the reserve power in the real-time operation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill -\Delta R·\left(\Delta S_{N_j}·\gamma ·P_t^{AvailCap}\right)\le P_{t,j}^{Up}-\Delta S_{N_j}·\gamma ·P_t^{AvailCap}\le \Delta R·\left(\Delta S_{N_j}·\gamma ·P_t^{AvailCap}\right);\forall t\in T,\forall j\in J,\forall r\in R.\end{array}\end{array}$$

(53)

Deployed down power from the reserve power in the real-time operation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill -\Delta R·\left(\Delta S_{N_j}·\gamma ·P_t^{AvailCap}\right)\le P_{t,j}^{Down}-\Delta S_{N_j}·\gamma ·P_t^{AvailCap}\le \Delta R·\left(\Delta S_{N_j}·\gamma ·P_t^{AvailCap}\right);\forall t\in T,\forall j\in J,\forall r\in R.\end{array}\end{array}$$

(54)

Actual power generation of renewable energy resources in the real-time operation:

$$\begin{array}{c}\hfill -\Delta R·\left({\overline{P}}_{t,j,r}^{RT}·u_{t,j,r}\right)\le P_{t,j,r}^{RT}-{\overline{P}}_{t,j,r}^{RT}·u_{t,j,r}\le \Delta R·\left({\overline{P}}_{t,j,r}^{RT}·u_{t,j,r}\right);\forall t\in T,\forall j\in J,\forall r\in R,\end{array}$$

(55)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill u_{t,j,r}=u_{t,j^{\prime },r}\forall t\in T,\forall j,j^{\prime }\in J,\forall r\in R.\end{array}\end{array}$$

(56)

• Formulation of the robust optimization problem:

To minimize the impact of uncertainty and maximize profit, the robust optimization problem comprises the main problem and sub-problem with the auxiliary variable, $\Gamma$. The main problem in (57) uses the profit in the day-ahead market and $\Gamma$ as decision variables, and its goal is to maximize the aggregator’s profit in the worst cases of the sub-problem. The sub-problem in (58) represents the profit of the real-time operation with uncertain parameters. The linear constraints of day-ahead scheduling and real-time operation on the objective function are expressed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Max\sum _{t=1}^{N_T}& ({\pi }_t^{DA}·\left(\sum _{s=1}^{N_s}\left(P_{t,N_j,s}^{DA,DCH}-P_{t,N_j,s}^{DA,CH}\right)+\sum _{r=1}^{N_r}{P}_{t,N_j,r}^{DA,Energy}\right)\hfill \\ \hfill & +{\pi }_t^{DARS}·\left(\sum _{s=1}^{N_s}\left(P_{t,N_j,s}^{RS,CH}+P_{t,N_j,s}^{RS,DCH}\right)+\sum _{r=1}^{N_r}{P}_{t,N_j,r}^{RS,RES}\right)\hfill \\ \hfill & -\sum _{s=1}^{N_s}\left({\rho }_s^{DCH}·P_{t,N_j,s}^{DA,DCH}+{\rho }_s^{CH}·P_{t,N_j,s}^{DA,CH}\right)-\sum _{r=1}^{N_r}\left({\rho }_r·P_{t,N_j,r}^{DA,Energy}\right))\hfill \\ \hfill & +\Gamma ,\hfill \end{array}\end{array}$$

(57)

s.t: (6)–(56),

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Gamma \le \sum _{t=1}^{N_T}\sum _{j=1}^{N_J}\Delta S_{N_j}·& ({\pi }_{t,j}^{RT}·(\sum _{s=1}^{N_s}\left(P_{t,j,s}^{Up,DCH}+P_{t,j,s}^{Down,DCH}-P_{t,j,s}^{Up,CH}-P_{t,j,s}^{Down,CH}\right)\hfill \\ & +\sum _{r=1}^{N_r}\left(P_{t,j,r}^{Up,RES}-P_{t,j,r}^{Down,RES}-P_{t,j,r}^{RES,imb}\right))\hfill \\ & +{\pi }_{t,j}^{RTRS}·\left(\sum _{s=1}^{N_s}\left(P_{t,j,s}^{Down,CH}-P_{t,j,s}^{Down,DCH}\right)+\sum _{r=1}^{N_r}\left(P_{t,j,r}^{Down,RES}\right)\right)\hfill \\ & -\sum _{s=1}^{N_s}\left({\rho }_s^{DCH}·\left(P_{t,j,s}^{Up,DCH}+P_{t,j,s}^{Down,DCH}\right)+{\rho }_s^{CH}·\left(P_{t,j,s}^{Up,CH}+P_{t,j,s}^{Down,CH}\right)\right)\hfill \\ & -\sum _{r=1}^{N_r}\left({\rho }_r·P_{t,j,r}^{Up,RES}\right)).\hfill \end{array}\end{array}$$

(58)

4. Case Study

4.1. Main Assumptions

To verify the effectiveness of the proposed robust optimization model considering the serving ratio, we implemented aggregated resources with a single wind turbine (wind), a 5 MW/30 MWh energy storage (BESS#1),and a 3 MW/18 MWh energy storage (BESS#2). The marginal cost of the wind turbine and BESS#1 and BESS#2 in charging/discharging modes were assumed to be 3 $/MW, 1 $/MW, and 0.8 $/MW, respectively. The expected wind power and price data (day-ahead, day-ahead reserve, real-time, and real-time reserve prices) are used based on the data for 24 January 2016, for the NYISO-west region, as shown in Figure 5 and Figure 6, respectively, [25]. Especially, the real-time market and real-time reserve prices are settled in five-minute time steps. The results of the proposed model were obtained using CPLEX optimizer, and the code and data for the optimization model can be found in [29].

4.2. Simulation Results

4.2.1. Case Description

We aim to validate the aggregator’s profit-maximization strategy as $\Delta R$ and $\gamma$ vary. Therefore, we compare the profit results for ($\Delta R$ = 0, 0.2, 0.4) and ($\gamma$ = 0, 0.2, 0.4, 0.6, 0.8, and 1).

Table 2. Description of case studies.

Case	Serving Ratio for Reserve Power ($\mathit{\gamma }$)	Variation Interval for Uncertain Parameters ($\mathbf{\Delta }\mathit{R}$)	Description
1	0, 0.2, 0.4, 0.6, 0.8, 1	0	Results without impact of uncertain parameters
2	0, 0.2, 0.4, 0.6, 0.8, 1	0.2	Results with variation intervals 20%
3	0, 0.2, 0.4, 0.6, 0.8, 1	0.4	Results with variation intervals 40%

shows the values of $\Delta R$ and $\gamma$ used in the case study. The case with $\Delta R$ = 0 is categorized as Case 1, that with $\Delta R$ = 0.2 as Case 2, and that with $\Delta R$ = 0.4 as Case 3; Cases 1 to 3 are subdivided into six cases according to the $\gamma$ value. The variation interval setting from Case 1 to Case 3 indicates the level of uncertainty the aggregator encounters. Due to the robust optimization characteristic of representing the results for the worst-case scenario of uncertain parameters, a large variation interval value allows for the consideration of various worst-case scenarios in the optimization problem. Therefore, we assume that the variation interval is set to 0%, 20%, and 40%. We could have set a variation interval of more than 40%, but we believe that it is uncommon to encounter a day-ahead forecast error exceeding 40%. Therefore, we chose to set the maximum value for the variation interval at 40%. The variation interval is 0% in Case 1, the error could be 0%. In this case, the aggregator is not subject to the uncertainty of renewable energy and the delivered output in the real-time operations. When there are no uncertainties, the optimization results, excluding Equations (53)–(55), will be described.

With a 20% or 40% variation interval in Case 2 or Case 3, the renewable forecast error could be equal to or less than 20% or 40%. Additionally, the error in providing deployed up/down power in the real-time operations could be equal to or less than 20% or 40%. When uncertainties are considered, the optimization results, including Equations (53)–(55), will be represented.

Along with the classification from Case 1 to Case 3, divided by variation interval, subcases for each case are categorized based on serving ratio. The serving ratio is utilized to calculate the range of capacity available for providing as reserve power, as shown in Figure 4. Six subcases are established by discretely adjusting the serving ratio, with each value increasing by 20%. These subcases can be expressed as Case 1-1 to Case 1-6, Case 2-1 to Case 2-6, and Case 3-1 to Case 3-6. A serving ratio of 0 means that the aggregator can only participate in the energy bidding in the day-ahead market, while a serving ratio of 1 indicates that the whole range of aggregator’s available capacity can be used as the reserve power for deployment in the real-time operations.

4.2.2. Case 1: Results without Considering Uncertain Parameters (0% Variation Interval)

Table 3. Profit results for Case 1 (0% variation interval for uncertain parameters).

Case	Variation Interval for  Uncertain Parameters	Serving Ratio for  Reserve Power	Day-Ahead Profit			Real-Time Profit			Total Profit
			BESS#1	BESS#2	Wind	BESS#1	BESS#2	Wind
1-1	0	0	217.1	138.6	1651.6	0	0	0	2007.4
1-2		0.2	20.9	307.8	1497.7	529.1	37.4	172.2	2565.1
1-3		0.4	443.2	77.1	1433.2	480.7	430.9	299.6	3164.6
1-4		0.6	425.7	340.3	1161.1	505.8	242.9	808.7	3484.6
1-5		0.8	425.7	340.3	1161.1	505.8	242.9	808.7	3484.6
1-6		1	425.7	340.3	1161.1	505.8	242.9	808.7	3484.6

shows the profit results for Case 1. Evidently, as the serving ratio increases from 0 to 0.6, the total and real-time profits increase, while the day-ahead wind profit decreases and real-time wind profit increases. This is because, as the serving ratio increases, the amount of wind power utilized in the day-ahead energy market decreases, and the wind power in the day-ahead reserve market and real-time operation increases. The profit per BESS in the day-ahead markets and real-time operation by serving ratio varies depending on the profit-maximization objective.

If a serving ratio is 0.6 or higher, the profit results for the BESSs and wind power are equal. This means that the optimal strategy is to use below 60% of the sum of the hourly forecast power of wind turbine and maximum power of BESS as reserve power.

The aggregator’s profit is related to the amount of BESS charging/discharging power and wind power. Figure 7 shows the day-ahead energy schedule, Figure 8 shows the day-ahead reserve schedule, Figure 9 shows the deployed power in the real-time up operation, and Figure 10 shows the deployed power in the real-time down operation. In Figure 7, Figure 9 and Figure 10, BESS discharging power and wind power are represented by positive values, and BESS charging power is represented by negative values. In Figure 8, all the reserve values are shown as positive values because the reserves in the day-ahead operation are represented to have the potential to increase or decrease power.

Evidently, the BESS charge/discharge and wind schedules are price-sensitive. They respond more to higher day-ahead and real-time prices than to lower day-ahead and real-time reserve prices. In addition, in most cases in Case 1, charging power occurs at 7–8, 10, 13–16, and 22–23 h when prices are low, and discharging power occurs at 0–1, 11–12, and 17–21 h when prices are high. At 2–6 and 9 h, the charging/discharging strategies vary depending on the profit-maximization objective.

In Case 1-1, only the BESS charging/discharging power and wind power in the day-ahead energy market are shown. This is because the serving ratio is 0; thus, the reserve power is not utilized. Consequently, the deployed power in the real-time operation is not utilized. Accordingly, only the BESS and wind profits for the day-ahead market are shown, and the real-time profit is zero. This implies that it is not ideal for the aggregator to hold no reserves in situations where joint electricity-market participation is possible.

The results from Case 1–2 to Case 1–6 show that different strategies for BESS charging and discharging modes are used. With the participation in the day-ahead reserve and real-time markets, the amount of charging and discharging power in the day-ahead energy market tends to decrease as the serving ratio increases. Simultaneously, the amount of charging and discharging power in the day-ahead reserve market and real-time operation increases. The increase in the amount of day-ahead reserve charging/discharging power is because it can be settled at day-ahead reserve prices. Also, deployed charging/discharging power in the real-time operation increases. This means that more profit can be earned by utilizing day-ahead energy as day-ahead reserve and real-time operation, depending on the day-ahead and real-time market prices. The discharging power in the day-ahead reserve market is similar to the deployed discharging power in the real-time up operation, and the charging power in the day-ahead reserve market is similar to the deployed charging power in the real-time down operation. The deployed charging power in the real-time up operation and discharging power in the real-time down operation are rarely utilized. The wind power utilized in the real-time operation, along with the day-ahead energy and reserve markets, helps increase the aggregator’s profit; thus, the amount of wind power appeared at almost all times in the energy and reserve markets. In addition, to maximize profit, the amounts of wind power in the real-time up operation and the day-ahead reserve market have the same value. The wind power utilized in the real-time down operation is zero all the time, as it does not contribute to the increase in profit from the difference between the real-time prices and day-ahead reserve prices.

4.2.3. Case 2: Results with Considering Uncertain Parameters (−20∼20% Variation Interval)

Table 4. Profit results for Case 2 (−20∼20% variation interval for uncertain parameter).

Case	Variation Interval for  Uncertain Parameters	Serving Ratio for  Reserve Power	Day-Ahead Profit			Real-Time Profit			Total Profit
			BESS#1	BESS#2	Wind	BESS#1	BESS#2	Wind
2-1	0.2	0	217.1	138.6	1981.0	0	0	0	2336.7
2-2		0.2	434.9	365.8	1884.8	5.5	−168.0	170.5	2693.4
2-3		0.4	755.0	342.5	1805.3	−175.9	−60.3	280.1	2946.7
2-4		0.6	943.6	453.8	1590.0	−409.7	−246.5	698.8	3030.1
2-5		0.8	1052.8	242.8	1610.3	−514.8	−55.8	710.2	3045.4
2-6		1	1183.6	74.7	1602.1	−616.6	87.5	742.1	3073.4

shows the profit results for Case 2. Clearly, the total and real-time wind profits increase as the serving ratio increases. Simultaneously, the day-ahead wind profit tends to decrease. Furthermore, when the day-ahead BESS profit increases, the real-time BESS profit tends to decrease, and vice versa, the real-time BESS profit tends to increase. Due to the limited deployed up/down power that can be utilized by applying Equations (53) and (54), it can be seen that the real-time BESS profit for Case 2 has a negative value.

For a serving ratio of 0 or 0.2, the total profit for Case 2 is higher than that for Case 1. For a serving ratio greater than 0.2, the total profit for Case 2 is lower than that for Case 1. The day-ahead BESS and wind profits for Case 2 are larger than those for Case 1, indicating larger day-ahead profits. However, the real-time BESS and wind profits for Case 2 are smaller than those for Case 1, indicating smaller real-time profits. Except for the total profit for Case 2-2, the total profits for Case 2-3 through 2–6 are smaller than those for Case 1-3 through 1–6. This is expected to be a result of the aggregator bidding conservatively in the day-ahead market rather than in the real-time operation due to the uncertainty of wind and deployed power in the real-time operation. The conservative bidding can be seen by the fact that the real-time profit for Case 2 is smaller than that for Case 1, or by the difference in the day-ahead profit.

The day-ahead energy schedule, day-ahead reserve schedule, deployed power in the real-time up operation, and deployed power in the real-time down operation for Case 2 are shown in Figure 11, Figure 12, Figure 13 and Figure 14, respectively. In Case 2, the discharging power occurs at 18:00 when the day-ahead price is the highest, and the charging power occurs at 23:00 when the day-ahead price is the lowest. During the rest of the day, the charging/discharging strategy varies depending on the profit-maximization objective of each BESS. In Case 1, the percentage of time when BESS#1 and BESS#2 charge/discharge simultaneously is high, whereas in Case 2, the percentage of time when BESS#1 and BESS#2 charge/discharge differently increased.

In Case 2, the charging power in the day-ahead energy market decreased and the discharging and wind power in the day-ahead energy market increased compared to those in Case 1. In addition, wind power in the day-ahead reserve market has decreased, and the charging/discharging power in the day-ahead reserve market has varied depending on the serving ratio. The charging power in the day-ahead energy market results in profit reduction, while the discharging and wind power in the day-ahead energy market increase the profit. Due to the low day-ahead reserve price, the charging/discharging and wind power in the day-ahead reserve market do not significantly impact the day-ahead profit.

The changes in charging/discharging and wind power in the real-time operation for Case 2 decrease the real-time profit. Compared to Case 1, the overall deployed power in the real-time operation has decreased, and the charging power in the real-time up operation, discharging power in the real-time down operation, and wind power in the real-time down operation increased. This means that variables that increase the profit have decreased and those that decrease the profit have increased. This also results in a negative real-time BESS profit.

In Case 2-1, constraints related to the charge/discharging power in the day-ahead energy market do not change, resulting in the same results as the charge/discharging power in the day-ahead energy market in Case 1-1. This also resulted in the same day-ahead profit per BESS. The difference between the two cases is the increase in the day-ahead wind profit in Case 2-1. This increase is due to the higher day-ahead wind power that can be utilized as a result of the increase in the value of $\Delta R$ in Equation (55).

From Case 2-2 to Case 2-6, the charging and discharging strategies vary and are not significantly affected by the serving ratio. This is because as the deployed up/down power in the real-time operation is small, the charging/discharging power in the real-time up/down operation that can be utilized is small. Consequently, the aggregator appears to have focused on bidding in the day-ahead energy and reserve markets.

4.2.4. Case 3: Results with Considering Uncertain Parameters (−40∼40% Variation Interval)

Table 5. Profit results for Case 3 (−40∼40% variation interval for uncertain parameter).

Case	Variation Interval for  Uncertain Parameters	Serving Ratio for  Reserve Power	Day-Ahead Profit			Real-Time Profit			Total Profit
			BESS#1	BESS#2	Wind	BESS#1	BESS#2	Wind
3-1	0.4	0	217.1	138.6	2298.0	0	0	0	2653.7
3-2		0.2	405.6	391.8	2200.0	54.5	−185.8	186.5	3052.6
3-3		0.4	757.7	387.7	2080.3	−171.1	−66.6	349.1	3337.0
3-4		0.6	892.4	479.5	1877.0	−324.0	−263.0	762.3	3424.1
3-5		0.8	884.6	442.0	1854.0	−301.2	−213.4	791.0	3456.9
3-6		1	1044.1	182.1	1849.3	−455.3	37.8	839.6	3497.5

shows the profit results for Case 3. As the serving ratio increases, the total, real-time, and real-time wind profits are found to increase, while the day-ahead wind profit is found to decrease. As in Case 2, the aggregator’s conservative bidding due to uncertain parameters is also seen in Case 3. However, the profits for Case 3 are generally larger than those for Case 2. This is likely due to the increased deployed up/down and wind power available for real-time operation because of the increased variation interval.

The day-ahead energy schedule, day-ahead reserve schedule, deployed power in the real-time up operation, and deployed power in the real-time down operation for Case 3 are shown in Figure 15, Figure 16, Figure 17 and Figure 18, respectively. In Case 3, the discharging power occurs at 17:00 and 18:00 when the day-ahead price is high, and the charging power occurs at 7:00 and 23:00 when the day-ahead price is low. For the rest of the day, the charging/discharging strategy varies depending on the profit-maximization objective of each BESS. In Case 3, the charging/discharging schedule is similar to that in Case 2.

Clearly, the day-ahead wind power in Case 3 increased compared to that in Case 2. In addition, the charging/discharging and wind power utilized as reserve in Case 3 increased. In most cases, the charging/discharging power in the real-time up/down operations in Case 3 decreased compared to those in Case 2, which can be understood as a decrease in the value of the profit-decreasing variable, resulting in an increase in the real-time BESS profit. Also, due to the increase in wind power in the real-time up operation, real-time wind profit has increased.

5. Discussion

In the case study, we analyzed the profit-maximization strategy of an aggregator that owns wind turbine and energy-storage systems under varying intervals of uncertain parameters and varying serving ratios.

When the serving ratio is zero (Cases 1-1, 2-1, and 3-1), only the day-ahead BESS and wind profits are shown, and the real-time profit is zero. As the variation interval increases, the day-ahead wind power that can be utilized increases, resulting in large day-ahead wind profits in the order of Case 1-1, 2-1, and 3-1; however, the profits are smaller than those in the case of utilizing reserve power, because the reserve power that can be utilized in the real-time operation is not secured. These cases can be thought of as representing the profits of an aggregator who can only participate in the energy market. The aggregator adjusts the charging and discharging power of energy storage according to the uncertainty of renewable energy to reduce the fluctuation in the power they can provide in the day-ahead market.

For non-zero serving ratios, real-time operations benefit from utilizing reserves in the real-time operations. As the serving ratio increases, the power used in the day-ahead energy market decreases, while that used in the day-ahead reserve market and real-time operation and the total profit increase. This suggests that aggregators can benefit both aggregators and system operators by utilizing day-ahead energy as day-ahead and real-time reserves. In particular, as the serving ratio increases, day-ahead wind profits tend to decrease and real-time wind profits tend to increase; this is because the amount of wind power utilized in the day-ahead energy market decreases and the amounts of wind power in the day-ahead reserve market and real-time operation increase. This can be explained by the fact that the profit from utilizing renewable energy in a real-time operation is greater than that from utilizing BESS charging and discharging power. In addition, in this case, the BESS charging/discharging strategies vary depending on the profit-maximization objective, resulting in different day-ahead and real-time BESSs profits.

Case 1 shows the results without considering the uncertain parameter, while Cases 2 and 3 show the results with the uncertain parameter. The day-ahead BESS and wind profits in Cases 2 and 3 are larger than those in Case 1, while the real-time BESS and wind profits are smaller than those in Case 1. This can be explained by the fact that due to the uncertainty of wind and deployed up/down power in the real-time operation, the aggregator focuses on bidding in the day-ahead market rather than utilizing power in the real-time operation. In other words, in Cases 2 and 3, the aggregator uses a strategy of seeking stable profits by focusing on bidding in the day-ahead energy and reserve markets. This strategy results in positive real-time BESS-specific profits in Case 1 and negative values in Cases 2 and 3. In Case 1, the absence of uncertainty leads the aggregator to maximize profits by properly utilizing the day-ahead market and real-time operation. In Cases 2 and 3, the aggregator increases the day-ahead BESS discharging power, day-ahead wind-power generation, and reserve charging/discharging power and decreases the day-ahead BESS charging power to obtain stable profits from the day-ahead market. This results in an increase in day-ahead BESS and wind profits. The real-time operation tends to charge power for use in the day-ahead market. As the profit-increasing variables decrease and the profit-decreasing variables increase, the real-time BESS profit appears negative. Comparing the profits for Cases 2 and 3, those for Case 3 are generally larger than those for Case 2 due to the increase in power available for wind and real-time operation as the variation interval increases.

It can thus be concluded that aggregators’ bidding strategy is closely related to price. Although the bidding strategies vary by the variation interval for uncertain parameters and serving ratio, we find that aggregators generally discharge at 17:00 and 18:00 when the day-ahead price is high and charge at 7:00 and 23:00 when the day-ahead price is low. They also respond more to higher day-ahead and real-time prices than to lower day-ahead and real-time reserve prices. It is believed that the price differential between the day-ahead and real-time markets will helps ensure adequate reserve capacity and mitigates the differences between the day-ahead market and real-time operation.

6. Conclusions

In this study, we proposed a robust optimization model that considers the serving ratio when the aggregator participates in the day-ahead market and real-time operation. The day-ahead market is assumed to comprise an energy market and a reserve market, and the reserve is determined by the sum of the forecast power of renewable energy resources and maximum power of energy storage multiplied by the serving ratio. The aggregator can participate in the real-time operations by increasing or decreasing power within the reserve range and receives the price in the real-time market. In addition, to consider the uncertainty in the real-time operation, the deployed up and down power in the real-time operations and the actual power generation of renewable-energy resources are configured as uncertain parameters, so that the values can be determined within the variation intervals. To minimize the impact of uncertainty and maximize the profit, the optimization problem comprises a main problem and a sub-problem represented by an auxiliary variable. The main problem is represented by the aggregator’s income/cost related to the day-ahead market and the sum of the auxiliary variables, and the sub-problem is represented by the aggregator’s income/cost related to the real-time operation. The optimization problem is modeled as a mixed-integer linear optimization problem with an objective function and linear constraints.

In the case study, the profit of an aggregator owning a wind turbine and two BESSs is analyzed according to the variation interval for uncertain parameters and the serving ratio. When the serving ratio is 0, the reserve power and deployed up/down power in the real-time operation are not utilized, and only the charging/discharging and wind power in the day-ahead energy market are utilized. When the serving ratio is non-zero, the benefit of real-time operation is shown by utilizing the reserve power and the deployed up/down power in a real-time operation. The BESS and wind profits in the day-ahead market with uncertain parameters are larger than those without them. On the other hand, the BESS and wind profits in the real-time operations are smaller than those without them. Due to the uncertainty, the aggregator uses a conservative bidding strategy in the day-ahead market rather than utilizing power for real-time operation. This bidding strategy also results in negative BESS profits in the real-time operation when the uncertain parameters are considered. Optimization results that consider volatile conditions due to uncertainty can provide aggregators with revenues that reflect changes in operating conditions between day-ahead and real-time operations. The increase in available reserve capacity, achieved by raising the serving ratio, can not only provide substantial revenues for aggregators but also be expected to offer system operators a significant reserve capacity to ensure the stable operation of the system. Furthermore, the price differential between the day-ahead and real-time markets helps ensure adequate reserve capacity and mitigates the differences between the day-ahead market and real-time operation.