The Optimal Energy Management of Virtual Power Plants by Considering Demand Response and Electric Vehicles

Abstract

This paper aims to explore Virtual Power Plants (VPPs) in combination with Demand Response (DR) concepts, integrating solar power generation, Electric Vehicle (EV) charging and discharging, and user loads to establish an optimal energy management scheduling system. Willingness curves for load curtailment are derived based on the consumption patterns of industrial, commercial, and residential users, enabling VPPs to design DR mechanisms under Time-of-Use (TOU), two-stage, and critical peak pricing periods. An energy management model for a VPP is developed by integrating DR, EV charging and discharging, and user loads. To solve this model and optimize economic benefits, this paper proposes an Improved Wolf Pack Search Algorithm (IWPSA). Based on the original Wolf Pack Search Algorithm (WPSA), the Improved Wolf Pack Search Algorithm (IWPSA) enhances the key behaviors of detection and encirclement. By reinforcing the attack strategy, the algorithm achieves better search performance and improved stability. IWPSA provides a parameter optimization mechanism with global search capability, enhancing searching efficiency and increasing the likelihood of finding optimal solutions. It is used to simulate and analyze the maximum profit of the VPP under various scenarios, such as different seasons, incentive prices, and DR periods. The verification analysis in this paper demonstrates that the proposed method can not only assist decision makers in improving the operation and scheduling of VPPs, but also serve as a valuable reference for system architecture planning and more effectively evaluating the performance of VPP operation management.

1. Introduction

Global warming has gradually attracted worldwide attention, and climate change caused by greenhouse gas (GHG) emissions has become a major concern for international organizations and governments [1]. The electric power sector contributes approximately one-third of total global GHG emissions, highlighting its critical role in addressing global warming. To reduce GHG emissions, the power sector is seeking more effective operational strategies. In recent years, peak electricity loads in Taiwan have been steadily increasing. Due to improvements in energy efficiency and national environmental policies, Taiwan is expected to rely primarily on renewable energy as its main power source in the future. Traditional centralized large-scale power generation will gradually transition to small-scale decentralized energy sources. Demand Side Management (DSM) will thus become a key factor in determining whether future energy systems can be managed effectively [2,3]. Since electricity cannot be easily stored and power transmission is constrained by line flow and safety reliability limits, the variability of renewable energy—often influenced by weather conditions—makes it difficult to ensure supply during peak demand. When the penetration of renewable energy becomes too high, power control and system management become increasingly complex. By integrating regional loads and Distributed Energy Resources into a Virtual Power Plant (VPP) through a centralized control center, VPPs are expected to play an important role in the future power grid [4,5].

Virtual Power Plants (VPPs) gather user groups operating under various Distributed Energy Resources (DERs) to monitor and coordinate electricity usage in real time. VPPs are expected to play a vital role in coordinating DERs to ensure reliable operations [6,7]. By aggregating the capacities of multiple DERs, a VPP can form a unified operating profile to participate in electricity markets. The introduction of VPP mechanisms can accelerate the transition away from thermal power dependence. Furthermore, VPPs can integrate flexible loads to support Demand Side Management (DSM), making them attractive to power utilities in deregulated electricity markets. Additionally, incorporating Electric Vehicle (EV) charging and swapping stations into VPPs can further enhance urban electricity consumption efficiency and achieve mutual backup capabilities [8,9]. Integrating EV charging and discharging into VPPs also enables better responsiveness to urban electricity demand. In the operation and dispatch of future power markets, VPPs are expected to play a key role while aligning with global low-carbon management objectives [10].

In recent years, numerous studies have focused on the scheduling and management strategies of Virtual Power Plants (VPPs), particularly on how different operational strategies affect VPP profitability. Reference [11] presents a VPP scheduling strategy from the perspective of a price maker, aiming to optimize bidding and maximize profits. Reference [12] proposes a VPP architecture for smart producers by integrating Distributed Generators (DGs), Energy Storage Systems (ESSs), and Demand Response (DR) within a local area to establish an energy trading platform. References [13,14,15] present a bi-level mathematical programming approach to identify the strategic bidding equilibrium of a VPP in a joint energy and regulation market in the presence of competitors. A Virtual Power Plant (VPP) can be established using a distributed power benefit-sharing platform to maximize the benefits for all participants [16]. Reference [17] proposes a novel framework for VPPs to address the challenges associated with renewable energy integration, particularly the uncertainty in wind and photovoltaic power outputs. The traditional method of generating profits solely from electricity sales is evolving; it is becoming a trend to incorporate Demand Response (DR) mechanisms and participate in VPP operation and dispatch strategies [18,19,20]. With the rapid growth of Electric Vehicles (EVs), the integration of solar carports with battery-powered vehicle charging and discharging capabilities further enhances the flexibility and overall effectiveness of a VPP power supply system [21]. The integration of Demand Response (DR) also increases VPP profits, offering advantages over traditional solar power plants. Reference [22] proposes a two-stage, bi-level dispatch optimization model for VPPs with EVs and DR based on a Stackelberg game framework. References [23,24] introduce a dual-objective energy trading model for VPPs, which considers the aggregated participation of EVs to enhance overall system efficiency. References [25,26] develop a Stackelberg game model incorporating deep reinforcement learning, in which VPPs and EV agents independently pursue their respective benefits. Reference [27] presents a two-layer optimization scheduling model, where the upper layer aims to maximize the profit of the VPP, while the lower layer focuses on minimizing the charging costs for EV users. Reference [28] develops an energy optimization framework to manage energy dispatch and leverage the flexibility of Electric Vehicles (EVs) to meet peak demand and enhance the profitability of VPPs. To improve the economic benefits of VPPs, Reference [29] proposes a three-stage scheduling strategy to coordinate EV participation. Reference [30] introduces a two-stage Vehicle-to-Grid (V2G) dispatching strategy to manage large-scale EV charging and discharging for optimal VPP operation. Reference [31] utilizes available EVs in parking lots to reduce peak loads in industrial centers, increase system profitability, and improve grid reliability. Reference [32] aims to identify and evaluate key uncertainties surrounding the deployment of EV batteries in VPPs, proposing strategic responses from an Environmental, Social, and Governance (ESG) perspective. The rapid growth of EVs has presented significant challenges to the flexible management and economic dispatch of VPPs. The participation of EVs in DSM has become a key research focus in recent years. Therefore, integrating ESS, EV charging/discharging, and DR to ensure the sustainable operation of VPPs will be one of the major challenges in the future.

This paper aims to explore VPPs combined with DR strategies by integrating solar power generation, EV charging/discharging stations, and user loads to establish an optimal energy management scheduling system. Based on the electricity consumption characteristics of industrial, commercial, and residential users, the load curtailment willingness curves of users [33] are derived to help VPPs plan DR mechanisms during TOU periods, two-stage periods, and critical peak periods. By considering DR strategies, incentive price, solar power generation, EV charging/discharging, and user loads, a VPP energy management model is established to maximize overall profit. This paper proposes the Improved Wolf Pack Search Algorithm (IWPSA) to solve the VPP energy management and dispatch model, aiming to accurately evaluate the VPP’s revenue potential. The IWPSA is based on the Wolf Pack Search Algorithm (WPSA) [34,35,36], and is designed to enhance two core behaviors: wolf detection and wolf siege. It strengthens the attack behavior of the wolves, improving both the search capability and the stability of the algorithm. IWPSA incorporates a parameter optimization mechanism with global search capabilities, enhancing computational efficiency and increasing the likelihood of finding optimal solutions. The algorithm is applied to simulate and analyze the maximum profit of a VPP under various scenarios, such as different seasons, incentive price, and DR periods. The verification analysis in this study is expected not only to assist decision makers in improving VPP operation and scheduling but also to provide valuable references for system architecture planning and enhance understanding of VPP operation management effectiveness.

2. Problem Formulation

This paper proposes an energy management system for VPPs aimed at maximizing their profit by considering DR and EV charging/discharging. VPPs integrate downstream users on behalf of power companies and sell electricity. When implementing DR, the VPP provides part of the rebate to users, allowing each user to propose the proportion of electricity consumption they are willing to reduce based on the incentive price. The VPP signs a DR contract with the user, offers an incentive price, and requires the user to provide a certain amount of load curtailment. The system architecture studied in this paper is shown in Figure 1.

2.1. The Willingness Curve of Load Curtailment for Users [37]

This paper categorizes users’ daily load curves into three types: industrial, commercial, and residential loads. The VPP integrates the load characteristics of various users with the incentive prices from DR contracts, as shown in Equations (1)–(3). When the incentive price is higher, users are more willing to curtail their loads. The total rebate received by users from the VPP can be determined based on the willingness curve of load curtailment.

$$DR{F}_{i,t}={\alpha }_i{\lambda }_{inc,i,t}^2+{\beta }_i{\lambda }_{inc,i,t}+c_i(\%)i=1,2,\dots ,n$$

(1)

$$DR{P}_{cur,i,t}=DR{F}_{i,t}\times DRloa{d}_{i,t}\left(\mathrm{kw}\right)i=1,2,\dots ,n$$

(2)

$$F{c}_{i,t}={\lambda }_{inc,i,t}\times DR{P}_{cur,i,t} (\mathrm{NT}\$)$$

(3)

$${Reb}_t={\sum }_{i=1}^{L}F{c}_{i,t}$$

(4)

$DR{F}_{i,t}$ is the willingness curve of load curtailment of the $i$-th user at time $t$. $DR{P}_{cur,i,t}$ is the agreed curtailed capacity of the $i$-th user at time $t$. $DRLoa{d}_{i,t}$ is the contract capacity of the $i$-th user at time $t$. ${\alpha }_i$, ${\beta }_i$, and $c_i$ are the parameters of the $DR{F}_{i,t}$ curve for the $i$-th user. $L$ is the number of users at time $t$. ${\lambda }_{inc,i}$ is the incentive price of the $i$-th user at time $t$. $F{c}_{i,t}$ is the rebate money of the $i$-th user at time $t$. ${Reb}_t$ is the total rebate money of users at time $t$.

2.2. The Model of the EVs

In the field of EVs, the energy for the electric motor that drives the vehicle is directly supplied by the vehicle’s battery. In our study, the EV model is described in Equation (5).

$${\mathit{P}}_{\mathit{b}\mathit{e}\mathit{v},\mathit{t}}={\mathit{P}}_{\mathit{b}}·{\mathit{S}}_{\mathit{n},\mathit{t}}$$

(5)

$P_{bev,\mathit{t}}$ is the total power of EVs at time $t$. $P_b$ is the power of an EV. $S_{n,\mathit{t}}$ is the number of EVs charged at a charging station at time $t$. The power output of the EVs is the difference between energy storages of two consecutive stages. Energy stored in a battery can be expressed as charging, that is,

$${\eta }_c{P}_{bev,t}\le Q_{s,max}$$

(6)

$$Q_{s,t+1}=Q_{s,t}+{{\eta }_{C}P}_{bev,t}$$

(7)

and discharging, that is,

$${\eta }_D{P}_{bev,t}\le Q_{s,t}$$

(8)

$$Q_{s,t+1}=Q_{s,t}-{{\eta }_{D}P}_{bev,t}$$

(9)

${\eta }_C/{\eta }_D$ is the charging/discharging co-efficient, respectively. $Q_{s,t}$ is the aggregated capacity of batteries at time $t$. $Q_{s,max}$ is the rated maximum energy storage.

2.3. The Profit of the VPP

The profit of the VPP is divided into three stages in the daily dispatch. The DR interval is implemented from the n-th period to the m-th period. The profit of the VPP for each stage can be described as follows:

(1)

The profit of the VPP before the start of DR is calculated as follows:

$$F_1={\displaystyle \sum _{t=1}^{n-1}\left({\lambda }_{vpp,t}{P}_{vpp,t}-{\lambda }_{pc,t}{P}_{pc,t}-{\lambda }_{gre,t}{P}_{gre,t}-{\lambda }_{bev,t}{P}_{bev,t}\right)}t=1,2,\dots ,n-1$$

(10)

(2)

The profit of the VPP during the DR interval is described as follows:

$$\begin{array}{c}{F}_2={\sum }_{t=n}^m[{\lambda }_{vpp,t}\left(P_{vpp,t}-P_{cur,t}\right)+{\lambda }_{DR,t}\left(P_{ref,t}-P_{pc,t}\right)-{Reb}_t-{\lambda }_{pc,t}{P}_{pc,t}-\hfill \\ {\lambda }_{gre,t}{P}_{gre,t}-{\lambda }_{bev,t}{P}_{bev,t}]\hfill \end{array}$$

(11)

(3)

The profit of the VPP after the end of DR is shown as follows:

$$F_3={\sum }_{t=m+1}^{24}\left({\lambda }_{vpp,t}{P}_{vpp,t}-{\lambda }_{pc,t}{P}_{pc,t}-{\lambda }_{gre,t}{P}_{gre,t}-{\lambda }_{bev,t}{P}_{bev,t}\right)$$

(12)

The total profit of the VPP in the daily dispatch is the sum of the profits from the three stages described above. The overall profit of the VPP, based on profit maximization, is shown in Equation (10).

$${Max.F}_{\mathrm{m}\mathrm{a}\mathrm{x}}=F_1+F_2+F_3$$

(13)

The relevant constraints are described as follows.

(1)

VPP energy balance constraint

$${P_{vpp,t}-P}_{pc,t}-P_{cur,t}-P_{bev,t}-P_{pv,t}-P_{gre,t}=0$$

(14)

(2)

Rebate constraints

$${\lambda }_{vpp,t}\le {\lambda }_{inc,t}\le {\lambda }_{DR,t}$$

(15)

(3)

The capacity of the battery

$$0\le \left(Q_s^{initial}+{\sum }_{t=1}^H{Q}_s^t\right)\le Q_{s,max}$$

(16)

$P_{vpp,t}/P_{pc,t}$: the load supplied by the VPP/utility at time $t$;

${\lambda }_{vpp,t}/{\lambda }_{pc,t}$: the price sold of the VPP/utility at time $t$;

${\lambda }_{inc,t}$: the incentive price of users by the VPP at time $t$;

${\lambda }_{DR,t}$: the incentive price of the VPP by the utility at time $t$;

${\lambda }_{gre,t}$/${\lambda }_{bev,t}$: the price sold of renewable energy/EVs at time $t$;

$P_{gre,t}/P_{bev,t}$: the generation of renewable energy/EVs at time $t$;

$P_{ref,t}$: the VPP’s historical electricity purchase for the 5 working days prior to time $t$.

$P_{cur,t}$: the load curtailed of users at time $t$;

$n$: the time when Demand Response starts;

$m$: the time when Demand Response ends;

$Q_{s,t}$: the capacity of the battery at time t;

$Q_s^{initial}$/$Q_{s,max}$: the initial/maximal capacity of the battery.

3. Methodology

The WPSA is a metaheuristic optimization algorithm inspired by the hunting behavior of wolves. The main behavioral strategies of WPAS include the following:

(1)

Searching and Roaming Behavioral Strategy: This strategy simulates the wolves’ random search for prey within their territory.

(2)

Calling and Chasing Behavioral Strategy: This strategy simulates the wolves’ coordinated chase and pursuit of identified prey.

(3)

Attacking and Capturing Behavioral Strategy: This strategy simulates the wolves’ collective attack on prey after it has been cornered.

The WPSA models the entire hunting process of a wolf pack using these three behavioral strategies. It begins by defining a search space with $i$ variables. The range of the $i$-th variable is between $mi{n}_i$ and ${max}_i$. The initial behavioral strategy is formulated as follows:

➢

Roaming Movement ($ste{p}_{a,i}$): The step size of the roaming movement is defined in Equation (17).

➢

Chasing Movement ($ste{p}_{b,i}$): The step size of the chasing movement is defined in Equation (18).

➢

Attacking Movement ($ste{p}_{c,i}$): The step size of the attacking movement is defined in Equation (19).

$$ste{p}_{a,i}={\displaystyle \frac{\left|{\mathit{max}}_i-{min}_i\right|}{\nu }}$$

(17)

$$ste{p}_{b,i}=2\times {\displaystyle \frac{\left|{\mathit{max}}_i-{min}_i\right|}{\nu }}$$

(18)

$$ste{p}_{c,i}={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{\left|{\mathit{max}}_i-{min}_i\right|}{\nu }}$$

(19)

$mi{n}_i$ and ${max}_i$ represent the lower and upper bounds of the search space, respectively. v is the mobility variable that enables the wolf pack to search for the optimal solution in a D-dimensional space.

However, there are two main shortcomings of the WPSA:

➢

The lack of necessary information exchange among the wolf pack may cause the wolves’ scouting behavior to become too dispersed, resulting in the algorithm moving away from the global optimum.

➢

The step size parameter of the attacking movement is constant. If it is set improperly, the convergence performance of the algorithm may degrade.

To address these issues, this paper proposes an Improved Wolf Pack Search Algorithm (IWPSA), which focuses on enhancing the reconnaissance behavior of scout wolves and the encircling strategy of the wolf pack. The proposed process is described as follows:

3.1. Searching and Roaming Behavioral Strategy

The odor concentration of the wolf king’s prey represents the current best solution ($Y_{\mathrm{lead}}$) in the solution space. The scout wolves continue to search for prey within the solution space. The position update of a scout wolf ($S_{i,d}$) is defined in Equation (20).

$$S_{i,d}^{\rho }=S_{i,d}+\mathit{sin}(2\pi \times {\displaystyle \frac{\rho }{h}})\times {step}_{a,i}\rho =1,2,\dots ,h\mathrm{and}d=1,2,\dots ,N$$

(20)

$N$ is the number of subgroups in the IWPSA. In this paper, N is set to 50. $S_{i,d}^{\rho }$ is the $\rho$-th directional solution of the $i$-th variable for the $d$-th scout wolf. The scout wolf $S_{i,d}$ explores a step in $h$ directions with each step taken using a step size referred to as ${step}_{a,i}$. The direction with the highest odor concentration is selected for movement, and the best solution state of $S_{i,d}$ is updated accordingly. When the odor concentration of prey by $S_{i,d}$ is $Y_{i,d}$, the corresponding solution is also denoted as $Y_{i,d}$. Since each wolf explores the space differently, the value of $h$ is randomly selected as an integer between $h_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $h_{\mathrm{m}\mathrm{a}\mathrm{x}}$ according to the size of the problem. In this study, $h_{\mathrm{m}\mathrm{i}\mathrm{n}}$ = 5 and $h_{\mathrm{m}\mathrm{a}\mathrm{x}}=20$ are set. A larger $h$ allows $S_{i,d}$ to search more precisely, but it also reduces the search speed.

In the IWPSA, a random value $r_1$ between 0 and 1 is generated for both the interactive roaming behavior and the interactive calling behavior. The number of maximal iterations $T_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is 100. The searching and roaming behavioral strategies of the wolves are updated as shown in Equations (21) and (22). This mechanism helps the wolf pack to better understand the main target, obtain global information, and maintain population diversity, thereby improving the wolves’ search capability.

$If{r}_1\ge 0.5$

$$S_{i,d}^{\rho }=S_{i,d}+\mathit{sin}(2\pi \times {\displaystyle \frac{\rho }{h}})\times {step}_{a,i}\rho =1,2,\dots ,h$$

(21)

else

$$S_{i,d}^{\rho }=S_{i,d}+\mathit{sin}(2\pi \times {\displaystyle \frac{\rho }{h}})\times \left|{S_{num}}_{i,d}^k-S_{i,d}\right|\rho =1,2,\dots ,h$$

(22)

end

During the search process, the prey odor concentration for the scout wolf is $Y_{i,d}$, and $Y_{lead}$ is the odor concentration perceived of prey by the wolf king. If $Y_{i,d}>Y_{lead}$, it indicates that the prey is closer to a scout wolf, and this wolf is most likely to capture the prey. Thus, $Y_{lead}$ is updated to $Y_{i,d}$. If $Y_{i,d}<Y_{lead}$, the scout wolf makes an autonomous decision to take a step in each of the $h$ directions. This movement is referred to as the roaming distance ${step}_{a,i}$, and the odor’s concentration is recorded. If the odor concentration of the prey is higher than the previous one, the wolf moves to the new position; otherwise, it returns to the original position.

3.2. Calling and Chasing Behavioral Strategy

The wolf pack members that respond to the call for chasing are referred to as fierce wolves. When the wolf king initiates a calling behavior by howling, it gathers the surrounding fierce wolves to rapidly move toward its location. The fierce wolves that hear the wolf king’s howl move quickly to the wolf king’s position using a relatively large chasing step. The position of a fierce wolf at iteration $k+1$ in the d-th dimension is given by Equation (23). The calculation of the distance between the position of a fierce wolf and that of the wolf king is shown in Equations (24) and (25).

$${S_{num}}_{i,d}^{k+1}={S_{num}}_{i,d}^k+{step}_{b,i}\times {\displaystyle \frac{G_i^k-{S_{num}}_{i,d}^{k+1}}{\left|G_i^k-{S_{num}}_{i,d}^k\right|}}d=1,2,\dots ,\mathrm{N}$$

(23)

$$D_{near}={\displaystyle \frac{1}{D\times \omega }}\times {\sum }_{i=1}^D\left|{\mathit{max}}_i-{min}_i\right|i=1,2,\dots ,D$$

(24)

$$D_{is}={\displaystyle \frac{1}{D}}\times {\sum }_{i=1}^D\left|G_i^k-{S_{num}}_{i,d}^k\right|i=1,2,\dots ,D$$

(25)

$G_i^k$ represents the position of the wolf king in the $i$-th dimension at the $k$-th generation. ${S_{num}}_{i,d}^k$ denotes the current position of the fierce wolf in the i-th dimension at generation k. Equation (22) shows that the fierce wolves gradually gather around the wolf king, reflecting the leadership of the wolf king over the pack. $D_{near}$ is the distance between the prey and the fierce wolf, while $D_{is}$ is the distance between the fierce wolf and the wolf king. $\omega$ is a distance-weighting factor. A larger $\omega$ accelerates the convergence of the algorithm but may reduce the wolves’ ability to explore the search space. The calling and chasing behaviors reflect the wolf pack’s mechanism for information exchange and sharing. During the search process, other individuals “follow” and “respond” to superior members of the group, demonstrating the sociality and intelligence embedded in the algorithm.

During the chasing process, the odor concentration of prey for the fierce wolf is $Y_{num,i}>Y_{lead}$, this fierce wolf becomes the new wolf king. Thus, $Y_{lead}$ is updated to $Y_{{num}_i}$. If $Y_{{num}_i}<Y_{lead}$, the fierce wolf continues to chase until the distance between it and the wolf king $G_i^k$ is $D_{is}<D_{near}$. At this point, the fierce wolf joins the encircling and capturing behavior. If the distance between the wolf king and the fierce wolf satisfies $D_{is}>D_{near}$, the calling behavior continues.

3.3. Attacking and Capturing Behavioral Strategy

After the fierce wolf approaches the prey through chasing, it begins to attack. The fierce wolves transform into an attacking wolf pack and capture the prey, as shown in Equations (26) and (27).

$$S{x}_{i,d}^k={S_{num}}_{i,d}^{k+1}$$

(26)

$$S{x}_{i,d}^{k+1}={Sx}_{i,d}^k+Rand\times {step}_{c,i}\times |G_i^k-{Sx}_{i,d}^k|$$

(27)

$Rand$ is a random number uniformly distributed between −1 and 1. In the IWPSA, a random value $r_2$ between 0 and 1 is generated for the interactive roaming and interactive calling behavior. The attacking and capturing behavioral strategy is renewed as Equations (28) and (29). This allows the wolf pack to adjust their encircling distance based on their proximity to the prey, thereby enhancing the algorithm’s ability to focus on the main target.

$If{r}_2\le 0.5$

$$S{x}_{i,d}^{k+1}=S{x}_{i,d}^k+Rand\times {step}_{c,i}\times \left|G_i^k-S{x}_{i,d}^k\right|$$

(28)

else

$$S{x}_{i,d}^{k+1}=S{x}_{i,d}^k+\lambda \times {step}_{c,i}\times {\displaystyle \frac{1}{2}}\left(rand\left|G_i^k-S{x}_{i,d}^k\right|+rand\left|G_i^k-Snu{m}_{i,d}^k\right|\right)$$

(29)

end

${step}_{c,i}$ represents the movement distance of the encircling wolf pack $S{x}_{i,d}^k$ in the $D$-dimensional variable space during the encircling behavior. If the prey odor concentration perceived by the wolf pack $x_i$ after executing the encircling behavior is $Y{x}_i>Y_{lead}$, then the position of the encircling wolf pack $S{x}_{i,d}$ is updated, and $Y_{lead}$ is set to $Y{x}_i$. Otherwise, the position of the encircling wolf pack $S{x}_{i,d}$ remains unchanged.

3.4. Wolf King Generation Rule in the Wolf Pack

The wolf king is identified in the initial solution space as the wolf with the best objective function value. During subsequent iterations:

• Compare the current wolf with the best objective value with the previous generation’s wolf king.
• If the current wolf has a better objective value, update the wolf king’s position.
• If multiple wolves have the same optimal value, randomly select one to become the new wolf king.

The selected wolf king does not participate in the three intelligent behaviors but proceeds directly to the next objective function evaluation until it is replaced by a stronger wolf.

3.5. Wolf Pack Update Mechanism

The wolf pack’s prey distribution rule is applied after hunting. Instead of equal distribution, prey is allocated based on a merit-based reward and strength hierarchy. The strongest wolves—those that first discover and capture the prey—are given priority, followed by the weaker wolves. This harsh distribution may result in weaker wolves starving due to food scarcity. However, it ensures that the most capable wolves remain strong for future hunts, thereby sustaining the continuity and development of the wolf pack.

In the IWPSA, the update mechanism reflects this distribution principle. Wolves with the worst objective function values are eliminated—R wolves are removed and replaced with R newly generated random wolves. A larger R increases population diversity but may cause the algorithm to become overly stochastic. Conversely, a smaller R reduces diversity and weakens the algorithm’s ability to explore new regions of the solution space. Considering that, in real wolf hunts, the size and quantity of captured prey vary, resulting in different numbers of weaker wolves starving, the value of R in this study is determined as shown in Equation (30).

$$R=\left[{\displaystyle \frac{N}{2\times \beta }},{\displaystyle \frac{N}{\beta }}\right]$$

(30)

$\beta$ is the population update ratio factor. In this paper, $\beta$ is set to 5.

3.6. Solution Process

The complete process of this study is as follows: First, the IWPSA is used to calculate the minimum generation cost for the power company for the following day. Next, DR incentives are calculated based on the TOU period, two-stage period, and critical peak period. Downstream users of the VPP submit their acceptable incentive levels and load curtailment amounts, which are categorized into willingness curves for industrial users and residential/commercial users. The power generation from solar carports and the number of EVs are then estimated to assess their impact on supply and demand. Finally, the optimal DR strategy for the VPP is evaluated, and the IWPSA is applied to determine the maximum daily profit for the VPP, as illustrated in Figure 2.

4. Case Study

This study simulates the optimal profits of the VPP under different electricity pricing periods, including TOU period, two-stage period, and critical peak period, for both summer and non-summer months. Figure 3a and Figure 3b illustrate the willingness curve for load curtailment of industrial users and residential/commercial users, respectively. The DR incentive prices for users are calculated using these curves to determine the amount of load curtailment ($P_{cur,t}$). The time intervals of electricity pricing periods are described as follows:

• TOU period: 10:00–16:00;
• Two-stage period: 10:00–12:00 and 13:00–16:00;
• Critical peak period: 13:00–15:00.

4.1. The VPP’s Profits in Non-Summer

Figure 4 shows the VPP’s electricity purchase plan from different users on non-summer working days. The amount of load curtailment by each user is determined based on the DR incentive prices, as shown in Figure 4.

The incentive price of DR from utility is set at two times ${\lambda }_{DR,t}$ and four times ${\lambda }_{DR,t}$. The DR implementation for users is TOU period, two-stage period, and critical peak period. Each user performs load curtailment according to the implementation period and incentive price, as shown in

Table 1. The load curtailment of users during the different periods.

Incentive Price	Time (hour)	TOU Period (TWD)	Industrial Load (kW)	Residential/Commercial Load (kW)	Two-Stage Period (TWD)	Industrial Load (kW)	Residential/Commercial Load (kW)	Critical Peak Period (TWD)	Industrial Load (kW)	Residential/Commercial Load (kW)
2 times of ${\lambda }_{DR,t}$	10	4	126.55	163.28	4	126.55	163.28	-	0	0
	11	4	127.61	164.50	4	127.61	164.50	-	0	0
	12	4	124.38	168.54	-	0	0	-	0	0
	13	4	117.69	165.00	4	117.69	165.00	4	117.69	165.00
	14	4.01	125.98	158.23	4	125.11	157.68	4	125.11	157.68
	15	4.01	126.77	163.25	4	125.89	162.68	-	0	0
	16	4	127.37	165.94	4	127.37	165.94	-	0	0
4 times of ${\lambda }_{DR,t}$	10	7.88	491.50	475.63	7.96	500.61	484.45	-	0	0
	11	7.92	500.24	483.63	7.96	504.83	488.07	-	0	0
	12	7.92	487.57	495.51	-	0	0	-	0	0
	13	7.99	469.81	494.01	7.96	465.58	489.56	7.99	469.81	494.01
	14	8.03	503.92	476.35	7.96	494.92	467.84	7.99	499.42	472.09
	15	8.03	507.08	491.46	7.96	498.02	482.67	-	0	0
	16	7.96	503.88	492.34	7.96	503.88	492.34	-	0	0

TWD: New Taiwan dollar.

. As shown in Table 1, regardless of whether it is the TOU period, two-stage period, or critical peak period, a higher incentive price results in greater load curtailment.

As shown in Table 1, when the incentive price is set at two times ${\lambda }_{DR,t}$, the VPP offers users a DR incentive price of TWD 4, meeting the minimum load curtailed requirement. Industrial users reduce their load by approximately 5% of their electricity consumption, while residential/commercial users reduce theirs by about 10%, indicating a higher curtailment rate among residential/commercial users. When the incentive price is set to four times ${\lambda }_{DR,t}$, the DR incentive price increases, leading to a greater willingness to curtail loads. The load curtailment of industrial users reaches 19.81% of their electricity consumption, while that of residential/commercial users reaches 25.47%. Despite this, industrial users have a higher electricity consumption base, resulting in a relatively higher curtailed amount.

Table 2. The maximal daily profit of VPP in non-summer.

Incentive Price	TOU Period (TWD)	Two-Stage Period (TWD)	Critical Peak Period (TWD)
2 times of ${\lambda }_{DR,t}$	59,509.98	59,573.63	59,824.68
4 times of ${\lambda }_{DR,t}$	61,705.01	61,421.55	60,440.79

TWD: New Taiwan Dollar.

shows the maximal daily profit of VPP in non-summer. When the DR incentive price is set to two times ${\lambda }_{DR,t}$, the VPP achieves its highest daily profit under the critical peak period. The maximal daily profit of VPP is TWD 59,824.86. Similarly, when the DR incentive price is set to four times, the highest profit occurs under the TOU period, reaching TWD 61,705.01.

4.2. The VPP’s Profits in Summer

Figure 5 shows the VPP’s electricity purchase plan from different users on summer working days. The DR incentive prices for various users are illustrated in Figure 3. The amount of load curtailment by each user is determined based on the corresponding DR incentive price.

As shown in

Table 3. The load curtailment of users during the different period.

Incentive Price	Time (Hour)	TOU Period (TWD)	Industrial Load (kW)	Residential/Commercial Load (kW)	Two-Stage Period (TWD)	Industrial Load (kW)	Residential/Commercial Load (kW)	Critical Peak Period (TWD)	Industrial Load (kW)	Residential/Commercial Load (kW)
Two times ${\lambda }_{DR,t}$	10	4.12	150.40	183.48	4.27	160.61	189.89	-	0	0
	11	4.21	155.96	191.14	4.27	159.77	193.59	-	0	0
	12	4.23	155.01	193.12	-	0	0	-	0	0
	13	4.25	146.71	186.49	4.27	147.88	187.28	4.30	150.23	188.86
	14	4.34	160.72	186.22	4.27	155.77	183.13	4.30	158.24	184.67
	15	4.32	162.26	189.79	4.27	158.48	187.42	-	0	0
	16	4.37	164.22	198.09	4.27	156.74	193.19	-	0	0
Four times ${\lambda }_{DR,t}$	10	8.24	601.60	566.72	8.53	642.43	605.18	-	0	0
	11	8.42	623.85	602.29	8.53	639.08	616.99	-	0	0
	12	8.46	620.03	610.87	-	0	0	-	0	0
	13	8.50	586.83	592.14	8.53	591.53	596.88	8.60	600.93	606.36
	14	8.68	642.87	602.19	8.53	623.07	583.64	8.60	632.97	592.91
	15	8.64	649.04	611.56	8.53	633.93	597.33	-	0	0
	16	8.75	656.86	645.07	8.53	626.98	615.72	-	0	0

TWD: New Taiwan dollar.

, when the DR incentive price is set at two times ${\lambda }_{DR,t}$, industrial users reduce their load by approximately 5.66% of their electricity consumption, while residential/commercial users reduce theirs by 10.66%, which is higher than in the non-summer period. When the DR incentive price is increased to four times ${\lambda }_{DR,t}$ reaching approximately TWD 8, the load curtailment increases significantly. The average load curtailment for industrial users reaches 22.63% of their electricity consumption, while that of residential/commercial users reaches 33.95%, indicating a substantial reduction in peak load.

The maximum profit of the VPP is shown in

Table 4. The maximal daily profit of VPP in summer.

Incentive Price	TOU Period (TWD)	Two-Stage Period (TWD)	Critical Peak Period (TWD)
Two times ${\lambda }_{DR,t}$	68,609.04	68,652.72	68,876.66
Four times ${\lambda }_{DR,t}$	74,932.88	74,583.93	72,884.66

TWD: New Taiwan dollar.

. When the DR incentive price is set at two times ${\lambda }_{DR,t}$, the VPP achieves its highest profit during the critical peak period, with a daily profit of TWD 68,876.66. When the incentive price is increased to four times ${\lambda }_{DR,t}$, the highest profit occurs during the TOU period, with a daily profit of TWD 74,932.88. The combined daily load curtailment from industrial and residential/commercial users reaches 8611.91 kW, indicating significant economic benefits. Due to higher electricity consumption in summer, the relative profit is TWD 13,227.87 greater than that in the non-summer period.

4.3. The DR of VPP with Solar and EV Charging/Discharging

This study simulates a VPP incorporating solar power generation and EV solar carports. The solar power capacity is 200 kW, and the daily solar power generation for non-summer and summer periods is shown in Figure 6. This solar power is not included in the DR incentives provided by the power company. The EV setup consists of four battery-powered vehicles, each with a capacity of 50 kW, for a total charging and discharging capacity of 200 kW. The EVs are fully charged by 18:00, with the charging costs covered by the VPP, and DR incentives also provided by the VPP. The DR incentive price is set at four times ${\lambda }_{DR,t}$, and the maximum profit of the VPP under various DR modes is calculated.

Table 5. The daily profit of the VPP in non-summer.

Hour	Original Load (kW)	TOU Period (kW)	Two-Stage Period (kW)	Critical Peak Period (kW)
1	3830.82	957.71	957.71	957.71
2	3702.06	925.52	925.52	925.52
3	3522.72	880.68	880.68	880.68
4	3298.20	824.55	824.55	824.55
5	3307.74	826.94	826.94	826.94
6	3356.34	839.09	839.09	839.09
7	3452.10	732.70	732.70	732.70
8	4027.44	2214.67	2214.67	2214.67
9	4874.88	2797.68	3112.05	3112.05
10	5128.26	3575.30 *	3681.37 *	3329.65
11	5483.94	3825.02 *	3928.83 *	3575.12
12	5609.82	3945.63 *	3695.93 *	3695.93
13	5728.80	4069.10 *	4158.77	4070.75
14	6092.64	4636.71 *	4401.55 *	4651.00 *
15	6277.02	4656.80 *	4455.21 *	4693.87 *
16	6178.20	4225.48 *	4327.00*	3972.44
17	5864.04	3390.11	3390.11	3390.11
18	5504.88	3130.15	3130.15	3130.15
19	5296.38	3317.76	3317.76	3317.76
20	5489.40	3436.36	3436.36	3436.36
21	5519.40	3455.14	3455.14	3455.14
22	5329.80	3336.45	3336.45	3336.45
23	4736.28	1184.07	1184.07	1184.07
24	4171.32	1042.83	1042.83	1042.83
Total		62,226.45	62,255.45	61,395.53

* Represents the DR execution.

and

Table 6. The daily profit of the VPP in summer.

Hour	Original Load (kW)	TOU Period (kW)	Two-Stage Period (kW)	Critical Peak Period (kW)
1	4824.50	1321.91	1321.91	1321.91
2	4392.90	1203.65	1203.65	1203.65
3	4092.53	1121.35	1121.35	1121.35
4	3828.11	1048.90	1048.90	1048.90
5	3662.07	1003.41	1003.41	1003.41
6	3821.85	1047.19	1047.19	1047.19
7	4003.95	857.39	857.39	857.39
8	4465.55	2464.60	2464.60	2464.60
9	5211.55	2973.62	2973.62	2973.62
10	5497.35	3890.26 *	4079.66 *	3551.97
11	5914.13	4284.62 *	4435.47 *	3905.06
12	6163.95	4483.07 *	4096.57 *	4096.57
13	6312.90	4589.25 *	4722.45	4514.79
14	6776.78	5250.46 *	5001.75 *	5252.16 *
15	6924.60	5277.26 *	5079.29 *	5291.70 *
16	6729.53	4798.65 *	4874.00 *	4347.89
17	6500.63	3828.03	3828.03	3828.03
18	6615.23	3834.93	3834.93	3834.93
19	6472.65	4057.44	4057.44	4057.44
20	6923.25	4333.95	4333.95	4333.95
21	7125.08	4460.30	4460.30	4460.30
22	6545.07	4097.21	4097.21	4097.21
23	6006.07	1645.66	1645.66	1645.66
24	5450.69	1493.49	1493.49	1493.49
Total		73,366.61	73,082.22	71,513.18

* Represents the DR execution.

present the daily profit of the VPP during the TOU period, two-stage period, and critical peak period in both non-summer and summer seasons. EVs are charged from the PV station during 07:00–09:00 and 17:00–19:00, resulting in lower profits during these periods. DR is executed from 10:00 to 16:00 across all three pricing periods; therefore, the VPP’s profit remains the same during other non-DR execution periods, and the main profit from DR occurs between 10:00 and 16:00. During the critical peak period, the maximum profit occurs at 14:00–15:00 when EVs discharge electricity to the VPP. The TOU and two-stage periods involve more evenly distributed discharging during DR, whereas the critical peak period concentrates discharging between 13:00 and 15:00. Regardless of summer or non-summer, the TOU period yields slightly higher daily profits than the other periods due to the longer DR execution time.

4.4. The Influence of the Incentive Price

The higher DR incentive price from the utility impacts the VPP’s operations. This study uses two, four, and six times ${\lambda }_{DR,t}$ to simulate the impact on VPP profitability during the TOU, two-stage, and critical peak periods.

Table 7. The influence of the incentive price among the various periods.

Month	Incentive Price	Load Curtailment		The Profit of VPP (NT$)
		Industrial Load (kW)	Residential/Commercial Load (kW)	TOU Period	Two-Stage Period	Critical Peak Period
Non-summer	Two times ${\lambda }_{DR,t}$	5	10	59,509.98	59,573.63	59,824.68
	Four times ${\lambda }_{DR,t}$	19.82	29.73	61,705.01	61,421.55	60,440.79
	* Four times ${\lambda }_{DR,t}$	19.82	29.73	62,226.45	62,255.45	61,395.53
	* Six times ${\lambda }_{DR,t}$	44.57	59.55	70,780.91	69,601.06	63,873.04
Summer	Two times ${\lambda }_{DR,t}$	5.67	10	68,609.04	68,652.72	68,876.66
	Four times ${\lambda }_{DR,t}$	22.88	33.95	74,932.88	74,583.93	72,884.66
	* Four times ${\lambda }_{DR,t}$	22.88	33.95	73,366.61	73,082.22	71,513.18
	* Six times ${\lambda }_{DR,t}$	48.96	63.27	81,903.44	80,377.89	73,951.35

* Considering PV and EVs.

shows the influence of incentive prices across the various periods. As shown in Table 7, higher incentive multipliers lead to greater VPP profits and increased load curtailment. During the TOU period, the VPP’s profit is higher than that in other periods. In addition, the VPP achieves even greater profits when EVs and PVs are considered the higher incentive price of DR from utility, impacting the VPP’s operations.

Table 8. Comparison of WPSA and IWPSA.

	Algorithm	Incentive Price	The profit of VPP (NT$)
			TOU Period	Two-Stage Period	Critical Peak Period
Non-summer	WPSA	4 times ${\lambda }_{DR,t}$	62,055.25	61,838.17	61,293.58
		6 times ${\lambda }_{DR,t}$	70,278.62	68,953.73	63,516.39
	IWPSA	4 times ${\lambda }_{DR,t}$	62,226.45	62,255.45	61,395.53
		6 times ${\lambda }_{DR,t}$	70,780.91	69,601.06	63,873.04
Summer	WPSA	4 times ${\lambda }_{DR,t}$	72,683.06	72,945.24	71,151.52
		6 times ${\lambda }_{DR,t}$	81,808.91	79,898.99	73,527.57
	IWPSA	4 times ${\lambda }_{DR,t}$	73,366.61	73,082.22	71,513.18
		6 times ${\lambda }_{DR,t}$	81,903.44	80,377.89	73,951.35

TWD: New Taiwan dollar.

compares the IWPSA with the WPSA during the TOU period, two-stage period, and critical peak period. Regardless of whether it is summer or non-summer, higher incentive prices bring greater profits for the VPP. The results also demonstrate that the IWPSA has superior performance in finding better solutions.

Table 9. Comparisons of EP, GA, PSO, WPSA, and IWPSA.

	The Profit of VPP (TWD)
Algorithm	Summer	Non-Summer
EP	72,645.63	62,012.56
PSO	72,235.69	61,987.25
GA	72,123.56	61,912.47
WPSA	72,683.06	62,055.25
IWPSA	73,366.61	62,226.45

shows a comparison chart of using EP, GA, PSO, WPSA, and IWPSA in the simulation. This study uses four times ${\lambda }_{DR,t}$ to simulate the profit of the VPP during the TOU period. From Table 3, it shows that the IWPSA can reach better results than EP, GA, PSO, and WSA. It can be seen that the IWSA improves the searching performance and explores a more likely global optimum.

5. Conclusions

This study has two main focuses:

• Utilizing the Improved Wolf Pack Search Algorithm (IWPSA) to calculate the daily profit of a Virtual Power Plant (VPP) by considering Demand Response (DR) and Electric Vehicles (EVs). DR events are implemented for TOU, two-stage, and critical peak periods, and the relationship between incentive price multipliers and the amount of load curtailment is analyzed.
• After the power company announces DR incentives, the VPP integrates regional electricity purchasing and selling, calculates user load curtailment, and uses the IWPSA to determine the maximum daily profit of the VPP.

This study simulates scenarios for workdays during non-summer and summer months to obtain baseline values for DR incentives under different pricing periods. Furthermore, the IWPSA is used to analyze the optimal daily profit of the VPP under various incentive multipliers. To enhance optimization performance, the traditional WPSA is improved by refining its search and encircling behaviors. These improvements increase the algorithm’s ability to find the global optimum and enhance the wolf pack’s diversity and search efficiency. The proposed algorithm is applied to simulate the profits of a VPP that incorporates PVs and EVs, enabling the VPP to manage energy storage in vehicles and earn DR incentives. Finally, this study calculates and evaluates the maximum profit of the VPP under different incentive multipliers, demonstrating the effectiveness of the IWPSA in optimizing VPP daily profit. It is believed that the analysis and verification conducted in this study not only assist decision makers in improving VPP operation and scheduling, but also provide valuable references for system architecture planning and help stakeholders better understand the impacts of VPP operation management.

VPP operations also inherited uncertain factors from DERs in the scheduling process; it is necessary to understand the characteristics and correlations of various DERs. In this paper, an optimized operation of the VPP is to maximize profits and perform the DR in the real-time procedure. However, challenges came from the intermittent nature of the renewables, which are hardly predictable and are dependent on weather and other climate factors. The profit of the VPP may be at risk due to the uncertainties. Decision makers require comprehensive risk assessments, providing a full distribution of profitability outcomes before making a decision. While risk is expected in the forming and operations of VPPs, addressing the risk of VPP operation would be a key issue in our future study topic.