Study of Two-Stage Economic Optimization Operation of Virtual Power Plants Considering Uncertainty

Abstract

In a highly competitive electricity spot market, virtual power plants (VPPs) that aggregate dispersed resources face various uncertainties during market transactions. These uncertainties directly impact the economic benefits of VPPs. To address the uncertainties in the economic optimization of VPPs, scenario analysis is employed to transform the uncertainties of wind turbines (WTs), photovoltaic (PV) system outputs, and electricity prices into deterministic problems. The objective is to maximize the VPP’s profits in day-ahead and intra-day markets (real-time balancing market) by constructing an economic optimization decision model based on two-stage stochastic programming. Gas turbines and electric vehicles (EVs) are scheduled and traded in the day-ahead market, while flexible energy storage systems (ESS) are deployed in the real-time balancing market. Based on simulation analysis, under the uncertainty of WTs and PV system outputs, as well as electricity prices, the proposed model demonstrates that orderly charging of EVs in the day-ahead stage can increase the revenue of the VPP by 6.1%. Additionally, since the ESS can adjust the deviations in day-ahead bid output during the intra-day stage, the day-ahead bidding strategy becomes more proactive, resulting in an additional 3.1% increase in the VPP revenue. Overall, this model can enhance the total revenue of the VPP by 9.2%.

1. Introduction

With the global transition in energy structures and the deepening development of power markets, the electricity spot market has become increasingly significant as a crucial component for the efficient allocation of power resources and market-based transactions [1]. In the electricity spot market, distribution networks serve as the critical link between the generation side and the consumption side, with their operational status and efficiency directly impacting overall market performance and user experience. However, the widespread integration of distributed energy resources, such as WTs and PV systems, along with the diversity and uncertainty of user loads, poses substantial challenges to the secure, stable operation and economic dispatch of distribution networks [2].

VPPs emerge as an innovative energy management technology that aggregates and manages various distributed resources within the distribution network, forming a unified, optimizable virtual entity. This approach provides a novel solution to address uncertainties in distribution networks [3,4]. In the context of the electricity spot market, VPPs not only participate as independent market entities but also leverage their flexible dispatch and resource optimization capabilities to effectively mitigate market uncertainties and achieve economically efficient distribution network operations.

Research on VPPs in electricity spot market transactions has primarily focused on bidding strategies, classifying VPPs as either a “price taker” or “price maker”. When a VPP acts as a “price taker”, it does not influence the price formation process and typically formulates its trading strategy based on price forecasts, making it highly susceptible to price fluctuations. This approach is particularly suitable for VPPs on the distribution network side. Reference [5] proposes a bidding strategy to maximize the overall profit of a VPP by considering day-ahead market clearing prices, power forecasts, and output constraints of distributed wind turbines (WTs) and photovoltaic (PV) systems, along with network structure constraints. Reference [6] introduces a multi-time scale economic dispatch strategy for VPPs, encompassing day-ahead bidding and intra-day operations to enhance economic benefits. Reference [7] presents a two-stage stochastic programming model with risk constraints, addressing the optimal bidding strategy for VPPs in day-ahead, intra-day, and spinning reserve markets. Although the “price taker” strategy facilitates VPPs’ participation in market transactions, developing flexible and effective bidding strategies necessitates comprehensive consideration of market bidding rules, operational costs, competitor bids, and operational risks to maximize profits. Reference [8] incorporates real-time balancing prices and competitor bidding strategies, using conditional value-at-risk to measure bidding risk, and establishes a bi-level optimal bidding model to enhance VPP operational profits. Reference [9] introduces a multi-market model for optimal VPP bidding strategies, converting fixed prices to dynamic response prices to improve demand response efficacy, achieving a win–win for operators and internal consumers. Additionally, VPPs can engage in hydrogen and carbon trading markets, efficiently utilizing renewable energy to reduce carbon emissions and contribute to environmental sustainability [10,11].

A critical research topic in VPPs is formulating economic optimization operation strategies in the electricity spot market while considering uncertainties. Distribution network uncertainties mainly arise from the stochastic nature of renewable energy output, unpredictable user loads, and dynamic market price variations. These uncertainties not only increase the complexity of VPP operations but also impose higher requirements for economic optimization. Currently, there are three primary methods for addressing uncertainties in VPPs: fuzzy theory, robust optimization, and stochastic programming. Reference [12] employ fuzzy theory to systematically represent imprecise and uncertain information, balancing economic and risk factors in VPPs optimization scheduling decisions to enhance operational efficiency and adaptability. Fuzzy theory also improves short-term load forecasting accuracy in VPPs [13]. However, the complexity of VPP system models and multiple uncertainties limit the practical application efficacy of fuzzy optimization [14]. References [15,16] adopt robust optimization, converting models into robust optimization models using strong duality and robust optimization theories, ensuring stable VPP profits under worst-case scenarios. Reference [17] proposes a data-driven robust optimization model to mitigate multiple uncertainties and enhance decision-making, while reference [18] discusses the importance of robust scheduling methods for adapting to dynamic operating conditions in VPPs. In VPP market bidding strategies, reference [19] emphasizes the importance of considering energy storage costs and transmission congestion in robust bidding strategies to improve market competitiveness and adaptability. Reference [20] incorporates demand response mechanisms into VPPs based on robust optimization, proven to enhance operational flexibility and resilience to renewable energy fluctuations. However, references [21,22] indicate that robust optimization often yields overly conservative results, sacrificing some economic benefits and requiring significant computational resources to address multiple uncertainties. Stochastic programming allows for the consideration of uncertainties in renewable energy generation and market prices, enabling VPPs to make informed, robust trading decisions [23]. Reference [24] uses probability density functions of wind speed and solar irradiance distributions, employing scenario generation and reduction techniques to create representative scenarios, describing WT and PV system output uncertainties using typical scenario sets. References [8,25] develop stochastic programming models to determine optimal VPP bidding and pricing strategies, maximizing profits and optimizing dispatch under renewable energy production and demand response uncertainties. The integration of EVs with VPP operations presents new challenges and opportunities. Reference [26] employs stochastic programming to optimize VPPs’ capacity allocation, efficiently meeting EV charging demands, managing EV charging schedules, minimizing costs, and enhancing overall benefits. Recently, advancements in information processing technologies facilitated more accurate acquisition of uncertainty variable probability distributions, reducing computational loads while efficiently utilizing data, thus broadening the applicability of stochastic programming in addressing VPPs uncertainties.

In summary, traditional VPP optimization strategies often overlook the real-time and flexibility requirements of the electricity spot market, particularly when facing challenges posed by uncertain renewable energy output and uncoordinated EV charging. To improve the economic efficiency and stability of VPPs in the spot market, it is essential to propose new optimization strategies to better manage intra-day deviation penalty costs and resource scheduling uncertainties. Therefore, this paper proposes a two-stage optimization strategy for VPPs’ day-ahead and intra-day operations, considering intra-day deviation penalty costs. Specifically, this strategy dispatches gas turbines and EVs in the day-ahead market while deploying flexible ESS in the intra-day real-time balancing market. The proposed bidding strategy and optimization algorithm aim to increase day-ahead market profits, reduce instability from uncoordinated EV charging, and achieve a win–win situation for VPPs and EV users. By using ESS to smooth output deviations, the strategy enhances VPPs’ flexibility and participation enthusiasm in the intra-day market. This comprehensive approach considers multiple distributed resources and their uncertainties, improving overall economic efficiency and operational stability.

The main contributions of this research are as follows:

(1)

Optimized Economic Benefits: Significant enhancement of VPPs market revenue through the rational scheduling of EVs and ESS;

(2)

Improved System Stability: Reduction of power supply instability from uncoordinated EV charging and reliable operation through flexible energy storage dispatch;

(3)

Increased Market Participation: Lowering intra-day market penalty costs and boosting VPPs day-ahead bidding enthusiasm and flexibility, promoting efficient renewable energy utilization;

(4)

Promoted Energy Transition: Providing new theoretical and practical foundations for VPPs applications in the electricity spot market, advancing towards more sustainable and intelligent power systems.

Through this research, we aim to offer new theoretical and practical insights for the optimal operation of VPPs, facilitating their widespread application in power markets and driving energy transition and sustainable development.

The remainder of this paper is structured as follows: Section 2 introduces the VPP structure and electricity market transaction framework. Section 3 details the scenario analysis method. Section 4 presents the two-stage stochastic programming model for the VPP economic optimization. Section 5 provides computational analysis. Finally, Section 6 concludes this paper.

2. The VPP Composition and Electricity Market Trading Framework

2.1. Composition of the VPP

VPPs efficiently manage and utilize diverse energy sources within power systems by integrating traditional power plants, distributed renewable energy, ESS, and flexible loads such as EVs. This integration optimizes energy production and consumption, balancing supply and demand in real time [27]. Such a setup allows VPPs to autonomously schedule activities in the electricity spot market, enhancing their operational capabilities and economic benefits [28,29]. As the cost of energy storage decreases, VPPs can more effectively mitigate the risks associated with renewable energy integration by utilizing ESS and participating in electricity market transactions. This not only improves the overall stability of the power system but also enhances its economic viability. The structure of the VPP constructed in this paper is illustrated in Figure 1. It comprises various components such as WTs, PV systems, ESS, gas turbines, flexible loads, and EVs. As an integrated “source-load-storage” system, the VPP leverages advanced communication, measurement, and control technologies, as well as software systems, to aggregate resources. The VPP scheduling control center coordinates these diverse resource elements, enabling unified participation in grid dispatch and market transactions. This centralized management and optimization control of distributed energy sources, storage systems, controllable loads, and EVs mitigate the impact of uncertainties on the grid. The profitability of a VPP is twofold: externally, it participates in grid dispatch operations and electricity market transactions; internally, it profits from the coordinated and reciprocal scheduling of “source-load-storage” elements.

2.2. WT Power Generation Units

The Weibull distribution model for wind speed [30] is expressed by its probability density function (PDF) as follows:

$$\mathrm{F}(\mathrm{v})=1-{\mathrm{e}}^{-{\left({\displaystyle \frac{\mathrm{v}}{\mathrm{c}}}\right)}^{\mathrm{k}}}$$

(1)

$$\mathrm{f}(\mathrm{v})={\displaystyle \frac{\mathrm{k}}{\mathrm{c}}}{\left({\displaystyle \frac{\mathrm{v}}{\mathrm{c}}}\right)}^{\mathrm{k}-1}{\mathrm{e}}^{-{\left({\displaystyle \frac{\mathrm{v}}{\mathrm{c}}}\right)}^{\mathrm{k}}}$$

(2)

where $v$ represents wind speed, $k$ represents the shape parameter, and $c$ represents the scale parameter. The shape parameter and the scale parameter can be computed using the following expressions:

$$\mathrm{k}=(\frac{\mathsf{\sigma }}{\stackrel{\text{‒}}{\mathrm{v}}}{)}^{-1.086}(1\le \mathrm{k}\le 10)$$

(3)

$$\mathrm{c}={\displaystyle \frac{\stackrel{\text{‒}}{\mathrm{v}}}{\mathsf{\Gamma }(1+\frac{1}{\mathrm{k}})}}$$

(4)

where $\stackrel{\text{‒}}{v}$ represents the mean wind speed, $\mathsf{\sigma }$ represents the standard deviation of wind speed, and $\mathsf{\Gamma }\left(x\right)$ represents the Gamma function.

The expression relating the power output of a WT to wind speed is

$$P_W=\left\{\begin{array}{cc}0& \hspace{1em}v<v_{\mathrm{ci}},v\ge v_{\mathrm{co}}\\ k_{1}v+k_2& \hspace{1em}{v}_{\mathrm{ci}}<v\le v_r\\ P_r& \hspace{1em}{v}_r<v\le v_{\mathrm{co}}\end{array}\right.$$

(5)

$${\mathrm{k}}_1={\displaystyle \frac{{\mathrm{P}}_{\mathrm{r}}}{{\mathrm{v}}_{\mathrm{r}}-{\mathrm{v}}_{\mathrm{ci}}}}$$

(6)

$${\mathrm{k}}_2=-{\mathrm{k}}_1{\mathrm{v}}_{\mathrm{ci}}$$

(7)

where $P_W$ represents the output power generated by the WTs; $P_r$ represents the rated power of the WTs; $v$ represents the wind speed affecting the WTs; $v_{ci}$ and $v_{co},$ respectively, signify the cut-in and cut-out wind speeds of the WTs; and $v_r$ represents the rated wind speed of the WTs.

2.3. PV System Power Generation Units

It is generally accepted that solar irradiance follows a Beta distribution [31], with the PDF expressed as

$$f(\gamma )={\displaystyle \frac{\mathsf{\Gamma }(\alpha +\beta )}{\mathsf{\Gamma }\left(\alpha \right)+\mathsf{\Gamma }\left(\beta \right)}}\times ({\displaystyle \frac{\gamma }{{\gamma }_{max}}}{)}^{a-1}\times (1-{\displaystyle \frac{\gamma }{{\gamma }_{max}}}{)}^{b-1}$$

(8)

where $\gamma$ and ${\gamma }_{max}$ represent the solar irradiance and maximum solar irradiance, respectively. The parameters $\alpha$ and $\beta$ are the two parameters of the Beta distribution, which can be calculated using the following expressions:

$$\alpha =w\left[{\displaystyle \frac{w(1-w)}{{\sigma }^2}}-1\right]$$

(9)

$$\beta =(1-w)[{\displaystyle \frac{w(1-w)}{{\sigma }^2}}-1]$$

(10)

The expression relating the output power of a PV system to solar irradiance is

$$P_{PV}=\left\{\begin{array}{c}{P}_{pv}{\displaystyle \frac{\gamma }{{\gamma }_e}}\hspace{1em}0\text{≤}\gamma \le {\gamma }_e\\ P_{pv}\hspace{1em}\gamma \ge {\gamma }_e\end{array}\right.$$

(11)

where $\gamma$ represents the rated solar irradiance of the PV system, and $P_{pv}$ represents the rated power of the PV system.

2.4. Gas Turbine Units

The mathematical model for the gas turbine is expressed as

$$P_{MT,t}={\displaystyle \frac{{\eta }_{\mathrm{MT}}\cdot L_{\mathrm{NG}}\cdot V}{\mathsf{\Delta }t}}$$

(12)

$${\eta }_{\mathrm{MT}}=a\times (p_{\mathrm{MT}}{)}^3+b\times (p_{\mathrm{MT}}{)}^2+c\times \left(p_{\mathrm{MT}}\right)+d$$

(13)

where $P_{MT,t}$ represents the output power of the gas turbine during the time period t; ${\eta }_{\mathrm{MT}}$ is the efficiency of the gas turbine; $L_{\mathrm{NG}}$ represents the lower heating value of natural gas, which is taken as 9.7 kWh/m³ in this paper; $V$ represents the volume of natural gas consumed during time period t; and a, b, c, and d are the fitting coefficients for the gas turbine.

2.5. ESS

The state of charge expression for the charging and discharging of ESS is

$$SOC={\displaystyle \frac{E_{rate}-E_d}{E_{rate}}}\times 100\%$$

(14)

where SOC represents the state of charge of the ESS, $E_{rate}$ is the rated capacity of the ESS, and $E_d$ is the capacity already released by the ESS.

The state of charge expression for the ESS during charging in time period t is

$$S_t=(1-\delta ∆t)S_{t-1}+{\displaystyle \frac{P_{ESSchar,t}\mathsf{\Delta }t{\eta }_c}{E_{rate}}}$$

(15)

The state of charge expression for the ESS during discharging in time period t is

$$S_t=(1-\delta \mathsf{\Delta }t)S_{t-1}-{\displaystyle \frac{P_{ESSdis,t}\mathsf{\Delta }t}{E_{rate}{\eta }_d}}$$

(16)

where $S_t$ represents the state of charge of the ESS during time period t; $\delta$ is the self-discharge rate; $P_{ESSchar,t}$ and $P_{ESSdis,t}$ represent the charging and discharging power of the ESS during time period t, respectively; and ${\eta }_c$ and ${\eta }_d$ represent the charging and discharging efficiencies, respectively.

2.6. EVs Units

EVs have dual attributes as both sources and loads due to their ability to charge and discharge, making them a resource with significant regulatory potential. Therefore, this paper considers EVs as controllable resources within a VPP. After the VPP signs agreements with EV owners, the control center can directly manage the charging and discharging of EVs when they are connected to the grid, according to the optimization strategy.

2.6.1. EVs Travel and Idle Periods

The primary function of EVs in a VPP is to meet the daily travel needs of residents. Given this characteristic, this paper optimizes the charging and discharging of EVs during the period from when owners return home and connect to the grid to when they leave for work and disconnect from the grid. This optimization achieves temporal and spatial energy transfer. Through refined management of EV charging and discharging, not only are the travel needs of owners met and the storage characteristics of EVs effectively utilized, but the economic benefits of the VPP are also enhanced. Based on statistical data, the PDF for the time periods when EV owners connect and disconnect from the grid is expressed as follows [32]:

$$f_{\mathrm{dep}}(x)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\mathrm{dep}}}}\mathit{exp}\left[-{\displaystyle \frac{{\left(x-{\mu }_{\mathrm{dep}}\right)}^2}{2{\sigma }_{\mathrm{dep}}^2}}\right],& 0<x\le \left({\mu }_{\mathrm{dep}}+12\right)\\ {\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\mathrm{dep}}}}\mathit{exp}\left[-{\displaystyle \frac{{\left(x-24-{\mu }_{\mathrm{dep}}\right)}^2}{2{\sigma }_{\mathrm{dep}}^2}}\right],& \left({\mu }_{\mathrm{dep}}+12\right)<x\le 24\end{array}\right.$$

(17)

$$f_{\mathrm{arr}}(x)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\mathrm{arr}}}}\mathit{exp}\left[-{\displaystyle \frac{{\left(x+24-{\mu }_{\mathrm{arr}}\right)}^2}{2{\sigma }_{\mathrm{arr}}^2}}\right],& 0<x\le \left({\mu }_{\mathrm{arr}}-12\right)\\ {\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\mathrm{arr}}}}\mathit{exp}\left[-{\displaystyle \frac{{\left(x-{\mu }_{\mathrm{arr}}\right)}^2}{2{\sigma }_{\mathrm{arr}}^2}}\right],& \left({\mu }_{\mathrm{arr}}-12\right)<x\le 24\end{array}\right.$$

(18)

where x represents a particular time point within the 24 h period, ${\mu }_{\mathrm{dep}}$ and${\mu }_{\mathrm{arr}}$ are the expected values of the EVs disconnection and connection times, respectively; and ${\sigma }_{\mathrm{dep}}$ and ${\sigma }_{\mathrm{arr}}$ are the standard deviations of the EVs’ disconnection and connection times, respectively.

2.6.2. EVs’ Driving Distance

The PDF for the driving distance of EVs is expressed as [33]

$$L_c(d)={\displaystyle \frac{1}{d{\sigma }_c\sqrt{2\pi }}}\mathit{exp}\left[{\displaystyle \frac{{\left(\mathit{In}d-{\mu }_c\right)}^2}{2{\sigma }_c^2}}\right]$$

(19)

where $\mathrm{d}$ represents the driving distance of the EVs, ${\mu }_c$ represents the expected value (mean) of the driving distance, and ${\sigma }_c$ represents the standard deviation of the driving distance.

The relationship between the initial State of Charge (SOC) and the driving distance for EVs is typically expressed as

$$SO{C}_{start}=SO{C}_{end}-{\displaystyle \frac{d{P}_c}{E_{EV}}}$$

(20)

where $P_c$ represents the EVs’ energy consumption per kilometer, $E_{EV}$ represents the battery capacity of a single EV, $\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{e}\mathrm{n}\mathrm{d}}$ represents the state of charge (SOC) at the end of the previous charging session, while $\mathrm{S}\mathrm{O}{\mathrm{C}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{t}}$ represents the SOC at the start of the current charging session.

2.6.3. EV Battery Pack Charging and Discharging Model

The state of charge of the EVs during charging in time period t is

$$SO{C}_{EV,t}=SO{C}_{EV,t-1}+{\eta }_{EVchar}{P}_{EVchar,t}$$

(21)

The state of charge of the EVs during discharging in time period t is

$$SO{C}_{EV,t}=SO{C}_{EV,t}-{\eta }_{EVdis}{P}_{EVdis,t}$$

(22)

where $SO{C}_{EV,t}$represents the state of charge of the EVs’ battery in time period t; $P_{EVchar,t,}$ and $P_{EVdis,t}$ represent the charging or discharging power of the EVs during time period t; and ${\eta }_{EVchar}$ and ${\eta }_{EVdis}$ represent the efficiency of charging and discharging for EVs.

After EVs are integrated into the VPP, the control center manages their charging and discharging schedules. However, due to the large number of EVs connected to the VPP, individually calculating the charging and discharging schedules for each vehicle would be computationally intensive. Therefore, this paper uses historical data on EVs to obtain PDFs for their departure times from and driving distances to the grid. Subsequently, employing Latin hypercube sampling and the Ng, Jordan, and Weiss (NJW) spectral clustering method, EVs with similar travel characteristics are clustered together. EVs within the same cluster, characterized by similar travel patterns, are treated as a single scenario set, enabling unified scheduling of each EV scenario set.

2.7. The Bidding Rules and Process for VPPs Participating in Spot Market Trading

The process of VPPs participating in electricity spot market trading is illustrated in Figure 2 as follows:

(1)

Prior to the conclusion of day D in the day-ahead electricity market trading, VPPs forecast the generation capacity of its internal distributed energy resources, internal load demands, and market electricity prices for day D + 1. Based on these forecasts, VPPs optimize the operation of its internal distributed resources with the goal of maximizing operational revenue. Subsequently, VPPs submit their bidding strategy for day D + 1 to the day-ahead electricity trading center (Independent System Operator, ISO). After the bidding process, the day-ahead electricity market operator announces the market clearing price and the awarded bid quantity for VPPs. VPPs then schedule the generation according to the awarded quantity on day D + 1 to ensure completion of the agreed-upon electricity transactions.

(2)

On day D + 1, VPPs control each distributed resource according to the day-ahead awarded bid quantity. Due to the difficulty in accurately predicting distributed WTs and PV system power outputs, there is often a deviation between the actual and planned outputs during operation. Consequently, there exists a discrepancy between the actual output of VPPs and the awarded bid quantity. If the actual output exceeds the awarded bid quantity, VPPs must sell the excess electricity in the intra-day market at a price lower than the market price. Conversely, if the actual operational output is lower than the awarded bid quantity, VPPs need to purchase additional electricity in the real-time balancing market at a price higher than the market price.

3. Scenario Analysis Method

3.1. Scenario Generation and Reduction for WTs and PV System Power Output

In this paper, we address the uncertainties in WTs and PV system power output using stochastic programming, where scenario generation and reduction techniques are employed to create a set of typical scenarios for WTs and PV system power output. This approach transforms the uncertainty problem into a deterministic one, allowing for model-based solutions. As illustrated in Figure 3, we calculate shape parameters based on historical wind speed and solar irradiance data, construct PDFs using Weibull and Beta functions, and employ Latin hypercube sampling and Kantorovich distance reduction methods to generate numerous wind speed and solar irradiance scenarios [34]. Corresponding power output functions are then used to generate typical WTs and PV system power output scenarios, with assigned weights representing the probability of each scenario occurring. This method accurately reflects the WTs and PV system power characteristics of the region, using the expected value of the generated scenarios as the actual output during real-time operation.

3.2. EVs Scenario Clustering

Upon integrating EVs into the VPPs, the VPPs’ control centers manage the charging and discharging schedules of the EVs. Given the large number of EVs, individually calculating and solving the charging and discharging plans for each vehicle would result in a significant computational burden. Therefore, we use historical data on EVs to obtain PDFs for their grid connection and disconnection times and travel distances. We then apply Latin hypercube sampling and the NJW spectral clustering method [35] to cluster EVs with similar travel characteristics. Each cluster, representing EVs with similar characteristics, is treated as a single scenario set, enabling unified scheduling for each EV scenario set.

3.3. Scenario Generation and Reduction for Market Prices

VPPs adopt a “price-taker” bidding strategy. We utilize the K-means clustering method to analyze historical day-ahead and real-time market price data, resulting in a set of typical joint price scenarios for day-ahead and real-time market prices [36].

4. Economic Optimization Decision Model for VPPs Based on Two-Stage Stochastic Programming

4.1. Two-Stage Optimization Strategy for VPPs

This paper assumes that virtual power plants (VPPs) adopt a “price taker” strategy when participating in the electricity spot market, with their bid quantities prioritized for clearing in the market. The participation of VPPs in the electricity spot market involves two stages. The first stage is the day-ahead market, where VPPs optimize the operation strategies of WTs, PV systems, gas turbines, and EVs and make decisions on bidding quantities before the clarity of WTs and PV system power outputs and electricity prices. The second stage is the real-time market, where considering the uncertainty in actual WTs and PV system outputs during the day, VPPs utilize ESS to pre-adjust the deviations in WTs and PV system outputs for each time period, thereby minimizing deviation penalty costs. The specific process is illustrated in Figure 4.

4.2. Objective Function

The profit of VPPs comes from revenue in the day-ahead market and the income (or expenditure) from balancing output deviations in the intra-day market. Due to the uncertainty of WTs and PV system outputs and day-ahead and real-time electricity prices, this paper employs scenario analysis methods to characterize WTs and PV system outputs. The decision model transforms the objective function into maximizing expected revenue across all scenarios, expressed as

$${\widehat{R}}_{vpp}=m\alpha x{\sum }_{t=1}^T{\sum }_{\omega =1}^M{\pi }_{\omega }(R_{t,\omega }^{Da}+R_t^{EVchar}-C_t^{MT}-C_t^{EVdis}-C_t^L)+m\alpha x{\sum }_{t=1}^T{\sum }_{\omega =1}^M{\pi }_{\omega }{\sum }_{s=1}^{N_1}{\gamma }_s{\sum }_{\nu =1}^{N_2}{\gamma }_{\nu }(R_{t,s,\nu }^{Rt}-C_t^{ESS})$$

(23)

where $w$, $s,$ and $v$ represent the sets of scenarios for intra-day electricity prices, WTs, and PV system power outputs, respectively. ${\pi }_w$, ${\gamma }_s,$ and ${\gamma }_v$ represent the probabilities associated with each scenario set for electricity prices, WT power, and PV system power. $R_t^{Da}$ represents the revenue from VPPs’ participation in day-ahead electricity market sales, $R_t^{EVchar}$ represents revenue from EV charging within VPPs, and $R_{t,s,\nu }^{Rt}$ refers to the revenue or costs incurred by VPPs in real-time balancing markets. $C_t^{MT}$, $C_t^{EVdis}$, $C_t^{ESS}$, and $C_t^L$ represent the operational costs of gas turbines, subsidies for EV discharging, costs associated with ESS charging and discharging, and adjustments for flexible loads, respectively.

4.2.1. Day-Ahead Electricity Market Revenue

After optimizing the internal power output schedule, VPPs submit the bid quantities for each time period of the next day. Upon successful bidding, each unit generates power according to the planned schedule in real-time. VPPs’ bid quantities and total sales revenue in the day-ahead market can be expressed as follows:

$$P_t^{Da}=P_{W,t}^{Da}+P_{PV,t}^{Da}+P_{MT,t}+\sum _{i=1}^N{P}_{EVdis,i,t}-\sum _{i=1}^N{P}_{EVchar,i,t}-{P_L}_{,t}$$

(24)

$$R_t^{Da}={\lambda }_t^{Da}{P}_t^{Da}\mathsf{\Delta }t$$

(25)

where$P_{W,t}^{Da}$, $P_{PV,t}^{Da}$, $P_{MT,t}$, $P_{EVchar,i,t}$, $P_{EVdis,i,t}$, and ${P_L}_{,t}$ represent WT power generation forecasting, PV system power generation forecasting, power outputs of gas turbines, EV cluster charging power, EV cluster discharging power, and the load demand of VPPs during time period t, respectively. ${\lambda }_t^{Da}$ represents the market clearing price in the day-ahead market.

The revenue from EV charging:

$$R_t^{EVchar}={\lambda }_t^{EVchar}{\sum }_{i=1}^N{P}_{EVchar,i,t}\mathsf{\Delta }t$$

(26)

where ${\lambda }_t^{EVchar}$ represents the charging tariff for EVs.

4.2.2. Revenue or Penalty Costs from Real-Time Electricity Market Transactions

In the second stage, the intra-day market, deviations often exist between planned and actual WTs and PV system outputs (simulated scenarios generate actual outputs in this paper). To mitigate these output deviations, this paper employs ESS as balancing units:

$$P_{t,s,v}^{Rt}=(P_{W,t,s}^{Rt}-P_{W,t}^{Da})+(P_{PV,t,v}^{Rt}-P_{PV,t}^{Da})+P_{ESSdis,t,}-P_{ESSchar,t}$$

(27)

where $P_{W,t,s}^{Rt}$ and $P_{PV,t,v}^{Rt}$ represent the actual WTs and PV system output values composed of the scenario set.

$$R_{t,s,v}^{Rt}=(1-\theta ){\lambda }_t^{Da}\mathsf{\Delta }{P}_{t,s,v}^{+}-(1+\theta ){\lambda }_t^{Da}\mathsf{\Delta }{P}_{t,s,v}^{-}$$

(28)

$$\mathsf{\Delta }{P}_{t,s,v,}^{+}=\mathit{max}\left\{P_{t,s,v}^{Rt},0\right\}$$

(29)

$$\mathsf{\Delta }{P}_{t,s,v}^{-}=\mathit{max}\left\{-P_{t,s,v}^{Rt},0\right\}$$

(30)

The equation specifies that $\theta$ is the penalty factor, indicating that VPPs will sell below the market price and buy above it in the intra-day market; ${\lambda }_t^{Da}$ represents the day-ahead market price.

4.2.3. Gas Turbine Fuel Costs

The gas turbines within VPPs consume natural gas during their operation. Therefore, the fuel cost of the gas turbine is calculated as follows:

$$C_t^{MT}=c^{MT}\cdot {\displaystyle \frac{P_{MT,t}\cdot \mathsf{\Delta }t}{{\eta }_{\mathrm{MT}}\cdot L_{\mathrm{NG}}}}$$

(31)

where $c^{MT}$ represents the unit price of natural gas, ${\eta }_{\mathit{MT}}$, $L_{\mathrm{NG}}$, and $P_{\mathit{MT}}$ are as mentioned earlier.

4.2.4. The Cost of Subsidies for EVs

Frequent control of EV charging and discharging by VPPs can shorten battery lifespan. Therefore, VPPs need to subsidize the discharging of EVs. The cost of subsidizing EVs is

$$C_t^{EV}=c^{EVdis}{\sum }_{i=1}^N(P_{EVdis,i,t})\mathsf{\Delta }t$$

(32)

where $c^{EVdis}$ represents the unit compensation cost for VPPs to control the charging and discharging of EVs, and $P_{EVdis,i,t}$ represents the discharge power of the EVs cluster at time t.

4.2.5. Operating Cost of ESS

ESS in VPPs shorten their lifespan with each charge and discharge cycle. Therefore, there exists an operating cost during operation:

$$C_t^{ESS}=c^{ESS}(P_{ESSchar,t}+P_{ESSdis,t})\mathsf{\Delta }t$$

(33)

where $c^{ESS}$ represents the unit operating cost of ESS.

4.3. Constraints

4.3.1. Constraints on WTs and PV System Power Output

The constraints on WT and PV System power output are as shown in Equations (34) and (35).

$$0\le P_{W,t}^{Da}\le P_{W,t}^{max}$$

(34)

$$0\le P_{PV,t}^{Da}\le P_{PV,t}^{max}$$

(35)

where $P_{W,t}^{Da}$ and $P_{PV,t}^{Da}$ represent the actual output of WTs and PV system power during time period t, while $P_{W,t}^{max}$ and $P_{PV,t}^{max}$ represent the upper limits of WTs and PV system power output during time period t. The upper limits of WTs and PV system power output correspond to the forecasted output values for the day.

4.3.2. Constraints on ESS

The constraints on ESS are as shown in Equations (36)–(39).

$${\alpha }_t^c{P}_{ESSchar}^{min}\le P_{ESSchar,t}\le {\alpha }_t^c{P}_{ESSchar}^{max}$$

(36)

$${\alpha }_t^d{P}_{ESSdis}^{min}\le P_{ESSdis,t}\le {\alpha }_t^d{P}_{ESSdis}^{max}$$

(37)

$$SO{C}_{ESS}^{min}\le SO{C}_t\le SO{C}_{ESS}^{max}$$

(38)

$${\alpha }_t^c+{\alpha }_t^d\le 1$$

(39)

where $P_{ESSchar}^{max}$ and $P_{ESSchar}^{min}$ represent the maximum and minimum charging power of the ESS, respectively; $P_{ESSdis}^{max}$ and $P_{ESSdis}^{min}$ represent the maximum and minimum discharging power of the ESS, respectively; $SO{C}_{ESS}^{max}$ and $SO{C}_{ESS}^{min}$ represent the maximum and minimum state of charge of the ESS, respectively; and ${\alpha }_t^c$ and ${\alpha }_t^d$ represent the charging and discharging efficiencies of the ESS, respectively.

4.3.3. Constraints on Gas Turbine Operation

The constraints on gas turbine operation are as shown in Equations (40) and (41).

$$P_{MT}^{\min }\le P_{MT,t}\le P_{MT}^{\max }$$

(40)

$$C_{MT}^{-}\le P_{\mathrm{MT},\mathrm{t}}\cdot P_{\mathrm{MT},t-1}\le C_{MT}^{+}$$

(41)

where $P_{MT}^{\max }$ and $P_{MT}^{\min }$ represent the maximum and minimum power generation of the gas turbine, respectively; and $C_{MT}^{+}$ and $C_{MT}^{-}$ represent the maximum power for the upward and downward ramping of the gas turbine, respectively.

4.3.4. Constraints on EVs

When VPPs control the charging and discharging of EVs according to scheduling strategies, EVs exhibit characteristics similar to ESS to external observers.

$$u_t^c{P}_{EVchar}^{min}\le P_{EVchar,i,t}\le u_t^c{P}_{EVchar}^{max}$$

(42)

$$u_t^d{P}_{EVdis}^{mi\mathrm{n}}{\le P_{EVdis,i,t}\le u}_t^d{P}_{EVdis}^{max}$$

(43)

$$SO{C}_{EV,i,t}=SO{C}_{EV,i,t-1}+{\eta }_{EVchar}{P}_{EVchar,i,t}-P_{EVdis,i,t}/{\eta }_{EVdis}$$

(44)

$$SO{C}_{EV,i}^{min}\le {SOC}_{EV,i,t}\le SO{C}_{EV,i}^{max}$$

(45)

$$u_t^d+u_t^c\le 1$$

(46)

where $SO{C}_{EV,i,t}$ represents the state of charge of EVs cluster i at time t; $SO{C}_{EV,i}^{max}$ and $SO{C}_{EV,i}^{min}$ represent the maximum and minimum state of charge of EVs cluster i, respectively; $P_{EVchar}^{\max }$, $P_{EVchar}^{min}$, $P_{EVdis}^{\max }$, and $P_{EVdis}^{min}$ represent the maximum and minimum charging and discharging power of EVs cluster i, respectively; $u_t^c$ and $u_t^d$ are binary variables representing the charging and discharging states of EVs cluster i at time t. These variables cannot both be 1 simultaneously in the same time period t.

5. Case Analysis

5.1. Basic Data

In this section, the VPP system aggregates distributed WTs and PV systems, gas turbines, EVs, ESS, and internal electrical loads. To verify the effectiveness of the two-stage economic optimization decision-making model for the VPP based on stochastic programming constructed in this chapter, the selected equipment parameters and price parameters for the case study are shown in

Table 1. Equipment parameters.

Equipment	Parameter	Value
Gas turbine	Maximum/minimum power generation (MW)	1.8
	Power generation efficiency	0.7
	Up and down climbing rate (MW/h)	0.5
EVs	Battery capacity of a single EV (MWh)	0.05
	Power consumption per kilometer (MWh)	0.00015
	Maximum and minimum State of Charge	0.9, 0.1
	Maximum charging/discharging power (WM)	0.01
	Minimum charging/discharging power (MW)	0
ESS (composition of lithium batteries)	Rated capacity (MWh)	4
	Maximum and minimum state of charge	0.9, 0.1
	Maximum charging power (MW)	0.4
	Minimum charging power (MW)	0

and

Table 2. Price parameters.

Parameter	Value (USD)
EV charging price (USD/MWh)	55.1436
EV discharge subsidy electricity price (USD/MWh)	68.9295
Operating cost of ESS charging and discharging (USD/MWh)	6.88
Natural gas (USD/m3)	0.4136
Penalty factor	0.2

.

The day-ahead forecast curves for the internal loads, PV power, and WT power of the VPP are shown in Figure 5.

In this paper, the models discussed in Section 3 and Section 4 were implemented using MATLAB 2020a. The Yalmip toolbox within MATLAB 2020a was employed, and the Gurobi solver was used to find solutions, with an average solving time of 125 s.

5.2. Scenario Generation and Clustering Results

5.2.1. WTs and PV System Scenario Generation and Reduction

In this paper, historical wind speed and solar irradiance data from a specific region were selected to construct 10 typical scenarios of WTs and PV system power output. These scenarios were utilized to simulate the actual intra-day output of WTs and PV system power. The generated WTs and PV system power output scenario set are illustrated in Figure 6.

Based on simulation results, the probabilities associated with each typical WT power output scenario are as follows: 0.09, 0.06, 0.10, 0.07, 0.18, 0.10, 0.17, 0.09, 0.05, and 0.09. Similarly, the probabilities for each typical PV system power output scenario are 0.08, 0.10, 0.08, 0.16, 0.06, 0.13, 0.12, 0.07, 0.13, and 0.07.

5.2.2. Clustering of Joint Electricity Price Scenarios

In this paper, K-means clustering was applied to historical electricity price data for day-ahead and intra-day markets. This clustering method yielded 10 typical joint electricity price scenarios, which are used to simulate the electricity prices when the VPP participates in market transactions. The joint electricity price scenario set is illustrated in Figure 7.

Based on simulation results, the probabilities associated with each typical electricity price scenario are as follows: 0.09, 0.06, 0.14, 0.06, 0.15, 0.05, 0.15, 0.10, 0.12, and 0.08.

5.2.3. Clustering Scenarios for Electric Vehicles

Considering the diverse range of EV models on the market, each with unique specifications, this paper simplifies the analysis by assuming all EVs belong to the same model type. The off-grid expected value for EVs is set to 7.45 with a standard deviation of 2.14, while the on-grid expected value is 17.30 with a standard deviation of 2.56. Using Latin hypercube sampling, numerical values for various characteristics of EVs were obtained. These values were then clustered into five clusters using the NJW spectral clustering method, with each cluster’s unified characteristics derived from the average values of all EVs within that cluster. The clustered characteristics of EVs are presented in

Table 3. Characteristics of EVs after clustering.

Cluster	Off-Grid Time (Hours/Minutes)	On-Grid Time (Hours/Minutes)	On-Grid SOC Status
1	3:52	14:46	0.714
2	5:55	16:26	0.645
3	7:34	17:35	0.626
4	9:12	20:24	0.668
5	11:13	21:41	0.639

.

5.3. Simulation Results Presentation and Analysis

5.3.1. First-Stage Day-Ahead Market Decision Results and Analysis

The internal dispatch strategy of the VPP in the day-ahead market, as depicted in Figure 8, leverages a “price-taker” bidding strategy, providing it with a competitive edge. Consequently, distributed WTs and PV system energy prioritize meeting internal load demands, with surplus electricity bid into the day-ahead market to generate revenue. During periods of low electricity prices, the gas turbine remains inactive due to higher generation costs compared to revenue from external sales, with WT power meeting internal demand. Conversely, during high-price periods, the gas turbine operates at full capacity as its generation costs are lower than revenue from external sales. Considering the gas turbine’s ramping constraints, it increases output continuously during 6–7 and 15–16 h and reduces output during 12–14 and 20–22 h. Additionally, EV discharge (negative) or charge (positive) during low-price periods from 1 to 5 and 22 to 24 h, and discharge during high-price periods from 17 to 21 h. This adjustment increases the VPP’s bid quantity during high-price periods, thereby enhancing its economic returns.

Figure 9 illustrates the correlation between the VPP’s bidding quantity in the day-ahead market, WTs and PV system power outputs, and clearing prices. During periods of higher WTs and PV system outputs, the VPP secures more successful bids in the day-ahead market. Similarly, higher clearing prices prompt increased bidding quantities. For instance, during 6–7, 13–14, and 20–21 h, despite lower clearing prices, the plant’s bids are higher due to greater WTs and PV system outputs. Conversely, during 8–11 and 17–19 h, when clearing prices are higher, the VPP also increases its bidding quantities. This reflects the plant’s ability to flexibly adjust bidding strategies based on market clearing prices and internal aggregation of distributed resources to maximize profits.

Different strategies for charging and discharging among the five clusters of EVs are depicted in Figure 10. The VPP optimally controls charging and discharging for each cluster to maximize benefits. Clusters 1, 2, 3, and 4 charge during periods of low market electricity prices while connected to the grid, meeting daily travel needs. During high-price grid-connected periods, they discharge to increase the VPP’s bid quantity in the day-ahead market, achieving a win–win situation for the plant operator and EV owners. Cluster 5, on the other hand, cannot exploit peak-demand pricing due to low electricity prices during grid-connected periods. Therefore, the VPP only regulates charging for this cluster and does not control discharging activities.

5.3.2. Second-Stage Intra-Day Market Decision Results and Analysis

The deviation in WTs and PV system power output is illustrated in Figure 11, depicting discrepancies between actual and planned outputs within the day. Variations in WTs and PV system output contribute to deviations between the VPP’s actual output during real-time operations and its forecasted bid quantities in the day-ahead market. Figure 12 shows the smoothed output deviations before and after adjustment. Positive deviations in the VPP’s output require selling surplus electricity in the real-time balancing market, while negative deviations necessitate purchasing electricity from the market. Before smoothing, actual WTs and PV system outputs during peak-price periods (8–11 and 17–19 h) are lower than planned outputs, resulting in the VPP needing to purchase additional electricity at higher prices in the real-time balancing market, incurring penalty costs due to deviations. To mitigate these costs, the VPP utilizes ESS to smooth out output deviations. After smoothing, the noticeable reduction in output deviations during 8–11 and 17–19 h indicates effective management of deviations through ESS.

The ESS within the VPP play a pivotal role in mitigating deviations in WTs and PV system output within the intra-day real-time balancing market. Beyond offsetting these deviations, they strategically plan charging and discharging activities based on the intra-day market prices and storage capacity. As depicted in Figure 12 and Figure 13, during low-priced periods, such as 1–2 h, the ESS procure electricity from the real-time balancing market at reduced rates for backup purposes. In periods with lower prices (3–5 h, 7 h, and 14–15 h) and higher actual WTs and PV system outputs compared to forecasts, the systems prioritize storing excess electricity for backup, later selling it when prices are higher. Conversely, during peak-price periods (8–11 h and 17–21 h), where penalties for shortfall in electricity supply are substantial, the ESS prioritize supplying the deficit and selling the remaining capacity at higher market prices, thereby maximizing revenue.

5.3.3. Economic Benefit Analysis

To validate the effectiveness of the proposed two-stage stochastic optimization model for the VPP operation, aiming to enhance its economic efficiency and stability, three scenarios are considered:

Scenario 1: The VPP aggregates WTs, PV systems, and gas turbines participating in both day-ahead and intra-day markets.

Scenario 2: The VPP aggregates WTs, PV systems, gas turbines, and EVs participating in both day-ahead and intra-day markets.

Scenario 3: The VPP aggregates WTs, PV systems, gas turbines, EVs participating in the day-ahead market, and ESS participating in the intra-day market.

Based on

Table 4. Day-ahead market benefits.

Scenario	Day-Ahead Market Revenue (USD)	EV Charging Revenue (USD)	Gas Turbine Fuel Cost (USD)	EV Discharging Subsidy Cost (USD)
one	7022.7401	0	1470.9693	0
two	7268.573	437.559	1470.9693	337.9958
three	7435.6843	437.559	1470.9693	337.9958

,

Table 5. Intra-day market benefits.

Scenario	Real-Time Balancing Market Revenue (USD)	ESS Operating Cost (USD)
one	144.5396	0
two	144.5396	0
three	219.994	52.8689

and

Table 6. Total benefits of the VPP.

Scenario	Day-Ahead Market Total Revenue (USD)	Real-Time Balancing Market Total Revenue (USD)	Total VPP Revenue (USD)
one	5551.7708	144.5396	5696.3104
two	5897.1669	144.5396	6041.7065
three	6064.2782	167.1251	6231.4033

, a comparison between Scenarios 1 and 2 reveals that incorporating EVs into the day-ahead market can significantly enhance the VPP earnings. By managing EV charging during off-peak electricity pricing periods to meet daily commuting needs and discharging surplus energy during peak pricing periods, the VPP’s revenue in the day-ahead electricity market increased from USD 7022.7401 to USD 7268.573, marking a gain of USD 245.8329. Although subsidizing EV charging imposes costs on vehicle owners, the VPP can strategically sell surplus energy during peak pricing, resulting in an overall revenue increase from USD 5696.3104 to USD 6041.7065—a rise of USD 345.3961, or approximately 6.1%. Moreover, implementing controlled EV charging and discharging schedules effectively mitigates the supply instability caused by unregulated EV charging.

Comparing Scenarios 2 and 3, where ESS are absent from the intra-day market, uncertainty in WTs and PV system outputs leads to penalty costs during periods of energy shortfall in the real-time balancing market. To minimize these penalties, Scenario 2 adopts a conservative strategy in the day-ahead market, reducing bid quantities. In contrast, Scenario 3 integrates ESS into the intra-day market, smoothing out output deviations and reducing penalty costs. Consequently, the VPP adopts a more proactive bidding strategy in the day-ahead market, increasing bid quantities and raising earnings from USD 7268.573 to USD 7435.6843—an increase of USD 167.1113. Additionally, by optimizing ESS charging and discharging strategies, earnings in the intra-day balancing market also improve, lifting the VPP’s total revenue from USD 6041.7065 to USD 6231.4033—a rise of USD 189.6967, or about 3.1%. Introducing ESS into the intra-day market enhances the VPP flexibility in day-ahead market decisions, thereby boosting system stability and overall operational profitability.

6. Conclusions

This paper introduces an innovative two-stage optimization strategy for the day-ahead and intra-day operation of VPPs, which incorporates intra-day deviation penalty costs. The strategy aims to achieve efficient VPP operation in the electricity market through precise scheduling of gas turbines, EVs, and flexible ESS. In the day-ahead stage, gas turbines and EVs are included in the scheduling framework. By leveraging the flexible charging and discharging capabilities of EVs, the strategy significantly enhances the VPP’s market revenue while mitigating the negative impact of uncoordinated EV charging on grid stability, thus creating a win–win situation for both VPP operators and EV users. In the intra-day market, ESS smooths out discrepancies between day-ahead forecasts and actual outputs. This approach successfully reduces the penalty costs faced by VPPs in the intra-day market. Simulation results indicate that through orderly management of EV charging in the day-ahead stage, the VPP’s revenue increased by 6.1%. In the intra-day stage, thanks to precise regulation by ESS, the strategy effectively mitigated deviations between day-ahead bids and actual outputs, resulting in an additional 3.1% revenue increase. Consequently, the overall revenue of the VPP improved by 9.2%. The proposed two-stage optimization strategy for VPPs provides precise scheduling of various resources, effectively addressing uncertainties in the electricity market. This significantly enhances the economic efficiency and stability of VPP operations, offering new insights and methods for the economic optimization of future VPPs.