Optimal Dispatch of a Virtual Power Plant Considering Distributed Energy Resources Under Uncertainty

Abstract

The varying characteristics of grid-connected energy resources necessitate a clear and effective approach for managing and scheduling generation units. Without proper control, high levels of renewable integration can pose challenges to optimal dispatch, especially as more generation sources, like wind and solar PV, are introduced. As a result, conventional power sources require an advanced management system, for instance, a virtual power plant (VPP), capable of accurately monitoring power supply and demand. This study thoroughly explores the dispatch of battery energy storage systems (BESSs) and diesel generators (DGs) through a distributionally robust joint chance-constrained optimization (DR-JCCO) framework utilizing the conditional value at risk (CVaR) and heuristic-X (H-X) algorithm, structured as a bilevel optimization problem. Furthermore, Binomial expansion (BE) is employed to linearize the model, enabling the assessment of BESS dispatch through a mathematical program with equilibrium constraints (MPECs). The findings confirm the effectiveness of the DRO-CVaR and H-X methods in dispatching grid network resources and BE under the MPEC framework.

1. Introduction

The global energy sector has embraced green energy initiatives aimed at reducing carbon emissions. This shift has accelerated the adoption of renewable energy technologies and has successfully promoted access to clean and affordable power worldwide. Notably, renewable energy has seen significant expansion, with 2023 alone accounting for 4.5%, equivalent to approximately 1310 TWh of the total global energy production [1].

Similarly, the findings in [2] project that renewable energy generation could rise from 28% in recent years to 68% by 2030, with a long-term projection of 91% by 2050. Given this substantial growth, implementing effective control and management strategies becomes vital to ensure the stable operation of the power grid. This is particularly important as renewable energy sources are inherently variable and less predictable, introducing uncertainties that complicate grid operation. Such variability may result in operational inefficiencies, such as the curtailment of excess energy, which leads to resource wastage. These challenges can be mitigated by applying robust control strategies, especially those that coordinate the dispatch of BESSs and DGs efficiently.

Research indicates that BESSs play a key role in maintaining grid stability, especially in frequency regulation [3]. This is further supported by a 42% improvement in cost-effectiveness between 2020 and 2030 [3]. Consequently, integrating BESSs with DGs offers a dependable solution to secure the grid against both internal and external disturbances, which typically should remain within defined thresholds. This study introduces a strategic control framework for generation units by applying a DR-JCCO technique in a bilevel structure to demonstrate the value of optimized dispatch for renewable integration.

Bilevel programming (BP) has emerged as a rapidly advancing field in optimization. Since the 1990s, it has gained recognition alongside other methods, such as stochastic and robust optimization [4]. BP problems are generally complex and demand specialized solution techniques, particularly when uncertainty is incorporated into the optimization model. It is well-suited for modeling interactions between two distinct decision-making agents, commonly referred to as the leader and the follower, which use a structured set of binding and non-binding constraints under uncertainty. The origin of bilevel modeling dates back to H. von Stackelberg in 1934 [5], and the mathematical application was later introduced by Bracken and McGill in 1973 [6]. The concept involves two agents making decisions that meet their respective objectives under clearly defined constraints.

Recently, BP has attracted significant attention, with various practical applications emerging. When uncertainties are integrated into single-level optimization, stochastic methods are generally adopted [7]. Bilevel formulations allow for a problem structure that can be addressed using established solvers. Mathematical approaches typically involve MPECs and equilibrium problems with equilibrium constraints (EPECs); the application of both these techniques depends on the problem structure. EPEC models are particularly useful in scenarios involving multiple VPPs [8], as they help determine fair pricing strategies for different market participants with varied bidding behaviors. In contrast, MPEC models are suited to single-level problems where bids from other entities are treated deterministically. In many cases, integer programming (IP) is used to solve MPEC problems [9], while binary decision strategies are also adopted to approximate nonlinear, continuous variables [10]. Mathematical formulation plays a pertinent role in accurately framing problems as robust, stochastic, or distributionally robust. In such cases, chance constraints incorporating probabilistic thresholds are used to confine system behaviors within acceptable limits.

Although the distributionally robust optimization problem (DROP) poses computational challenges due to its extreme conservative nature and complexity, which often results in static solutions that make it difficult to draw general conclusions, it has shown significant performance compared to robust or stochastic optimization in terms of minimal information loss and better out-of-sample performance. To address these limitations, approximation algorithms, such as conditional value at risk (CVaR), Bonferroni techniques, and moment-based methods are often applied. CVaR replaces the probability threshold in traditional chance constraints by estimating the expected losses beyond the value at risk (VaR). This approach is widely regarded as one of the most effective techniques for approximating joint chance-constrained problems. CVaR accounts for losses within a defined probability threshold and is also known as an expected shortfall. In [11], CVaR is used to assess system reliability under both stochastic and fuzzy uncertainties, helping to adjust the conservativeness of solutions when considering worst-case scenarios.

According to [12], the heuristic-X algorithm, termed as ALSO-X, outperforms the traditional CVaR method in solving chance-constrained problems across known distributions. It also performs well under type-∞ Wasserstein ambiguity sets, which use Wasserstein distance to quantify worst-case scenarios across several distributions. This method offers increased robustness by maintaining a fixed distance between distributions.

In addition, type-q Wasserstein ambiguity sets are another viable approach for evaluating worst-case deviations. Unlike the type-∞ formulation, this method only measures the distance between a nominal distribution and a group of alternatives, offering more flexibility and less conservativeness. The key challenge lies in ensuring robust solutions without compromising solution flexibility. The latter approach is often favored due to its reduced conservatism compared to type-∞ ambiguity sets. Studies, such as [13,14], have examined these approximation algorithms and concluded that, under certain conditions, the methods may yield sub-optimal results. This study advances the field by adopting these techniques and applying diverse solvers to derive optimal solutions under a unified framework in the energy sector.

This work is divided into the following sections.

Section 2 introduces the previous work and proposes approximation methods. Section 3 presents the solution methodology, which entails the mathematical modeling of DERs and the contribution of this research work. Section 4 details the numerical notation of the approximation algorithms using the existing mathematical equations. Section 5 presents the results followed by discussion, conclusion, future work, acknowledgements, conflicts of interest, and references.

2. Previous Work

The use of BP in the context of a VPP is a novel contribution of this study. In this framework, the VPP aims to maximize its profits in DM, while the second-level agent is considered to be the ISO, whose role is to clear the electricity market by minimizing operational costs, managing congestion, and ensuring that electricity is delivered to consumers at affordable prices.

Table 1. Existing bilevel models in the literature.

Ref #	Problem Formulation	Modeling Levels		Uncertainty	Solution Algorithms
		Upper Level	Lower Level
[9]	Bilevel	DN	MG	RO	ADMM
[15]	Bilevel	Energy hub	EVs	SP	—
[16]	Bilevel	Energy hub	Users	Two-stage RO	C&CG
[17]	Energy hub	Energy hub	Users and EVs	KL-based DRO	Crafted C&CG (linearization of USP)
[18]	Single level	—	—	KL-based DRO	Outer approximation
[19]	Single level	—	—	KL-based DRO	Benders decomposition
[20]	Single level	—	—	KL-based DRO	C&CG
[21]	Bilevel	DN	MG	—	Stackelberg game
[22]	Bilevel	DN	MG	RO	ADMM
[23]	Bilevel	DN	MG	—	Noncooperative game
[24]	Bilevel	DN	MG	—	KKT
[25]	Bilevel	DN	MG	DRO	KKT
[26]	Bilevel	TN	DN	IGDT	KKT
[27]	Bilevel	TN	DN	RO	Iterative method (two steps)
[28]	Single level	TN	MG	IGDT-SP	—
[29]	Bilevel	TN	MG	RO-SP	Dantzig–Wolfe
[30,31]	Single level	—	—	Moment-based DRO	Delayed constraint generation
Proposed	Bilevel VPP	Day-ahead market	VPP owners	DR-JCCO	CVaR&H-X

illustrates this interaction and also presents selected studies that have applied bilevel modeling.

3. Solution Methodology

This study applies a DR-JCCO framework to solve the optimization problem. The proposed methodology adopts a two-level structure known as the bilevel approach. In this setup, the ISO is responsible for minimizing the operational costs within the power system, formulated as a minimization objective. Meanwhile, the second-level objective is handled by the VPP operator, who seeks to maximize the overall social benefit derived from DERs.

Once the optimization stages are defined, joint chance constraints are introduced and maintained below a specified probability threshold. To handle this, DR-JCCO is employed, where the probability constraints are approximated using known probability distributions.

Both CVaR and H-X approaches will be implemented and compared in terms of their conservativeness and performance in addressing uncertainties within the grid network. Their effectiveness will be evaluated using wind power scenarios, particularly examining their impact on BESS charge and discharge performance. The binomial expansion (BE) technique is also used to develop the MPEC formulation for modeling the BESS dispatch, making it compatible with commercial solvers, such as Gurobi.

This study offers the following fundamental contributions:

• The adoption of a data-driven strategy to directly optimize the overall social welfare of market participants.
• The implementation of a bilevel optimization model where the first stage minimizes operational costs and the second stage maximizes social welfare.
• The proposal of a novel CVaR and H-X approximation techniques to coordinate DG commitment and BESS dispatch in the DR-JCCO model, an application which is considered to be the first of its kind.

The workflow for obtaining the results is illustrated in Figure 1.

3.1. Generators

DGs are conventional sources of generating energy, and they are modeled using the following equations:

$$P_{ij}+P_j^G=\sum _{k\in \pi \left(j\right)}{P}_{jk}+P_j^d$$

(1)

$$Q_{ij}+q_j^g=\sum _{k\in \pi \left(j\right)}{Q}_{jk}+q_j^d$$

(2)

$$V_j=V_j-{\displaystyle \frac{R_{ij}{P}_{ij}+X_{ij}{Q}_{ij}}{V_o}}$$

(3)

where $P_{it}^G$ [32] is generator power, which is quantified as $p_j^G$ and then multiplied by the incremental generation cost of conventional DGs (USD/MWh) $\gamma$, which is the cost incurred when DGs are utilized. $P_{ij}$ is the injected active power, while $Q_{ij}$ is the reactive power for the grid network (see Equation (2)). $P_{jk}$ is the active power, $P_j^d$ is the demand, $q_j^g$ is the generated reactive power, and $q_j^d$ is the required reactive power at bus $j$, as shown by Equation (3). $V_j$ is the bus $j$ voltage magnitude, $V_o$ is the slack bus, $R_{ij}$ is the resistance, and $X_{ij}$ is the reactance.

The following Equation (4) is the generator cost function:

$$C_{j^G,t}^G=a_j{\left(p_j^G\right)}^2+b_j{p}_j^G$$

(4)

where $C_{i^G,t}^G$ is the total generated power from the generator, which is approximated as shown by Equation (5) and then limited in Equation (6). $a_j$ and $b_j$ are both coefficients for the quadratic cost function, i.e., Equation (4).

$$\begin{gathered} p_j^G\sum _{j\in \pi \left(0\right)}{p}_{0j} \\ {C}_{i^G,t}^G\approx \gamma \sum _{{\Omega }_G}{P}_{it}^G \end{gathered}$$

(5)

$$0\le P_{it}^G\le P_{max }\forall t\in T$$

(6)

3.2. Wind Turbine

Equation (7) is associated with the Weibull distribution function, a standard equation usually applied to obtain WT power output. Equation (7) is as follows:

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

(7)

The correlation that exists between wind speed and the WT output $\left\{P_{wT,t}:t\in T\right\}$ can be represented by the following empirical Equation (8).

$$P_{WT}=\left\{\begin{array}{ll}{P}_{wT,r}& v_r\le v\le v_{co}\\ P_{wT,r}{\displaystyle \frac{v-v_{ci}}{v_r-v_{ci}}}& v_{ci}\le v\le v_r\\ 0& v<v_{ci} \mathrm{or} v>v_{co}\end{array}\right.$$

(8)

3.3. BESS

Equation (9) represents the battery cost function using $k_1$ and $k_2$ as coefficients; although they can be ignored for simplicity purposes, they are factored into the utility degradation function.

Equation (10) shows the minimum BESS discharging and charging power, while the rest of the equations define constraints that ensure that the safe operating conditions for the BESS are not violated.

$$\sum _{i^B\in \Omega BESS}^{BESS}{C}_{,t}^{BESS}=\sum _{t=1}^T{\lambda }_t{P}_t^{EC}-P_t^{ED}+k_1{d}_t+k_2{d}_t^2$$

(9)

$$P_{min}^{ED}\le P_t^{ED},P_t^{EC}\le P_{max}^{EC}$$

(10)

$$SO{C}_{it}=E_{i,t}^{BESS}/E_{i,R}^{BESS}$$

(11)

$$SO{C}_{min}\le SO{C}_{it}\le SO{C}_{max}$$

(12)

$$E_{i,t+1}^{BESS}=E_{i,t}^{BESS}-P_{i,t}^{BESS}\Delta t-E_{i,t}^{BESS}\eta \Delta t$$

(13)

$$P_{i,t}^{BESS}\le {\bar{P}}_i^{BESS,Dis}{\chi }_{i,t}^{Dis}$$

(14)

$$P_{i,t}^{BESS}\ge {\bar{P}}_i^{BESS,Chr}{\chi }_{i,t}^{Chr}$$

(15)

$${\chi }_{i,t}^{Dis}+{\chi }_{i,t}^{Chr}\le 1$$

(16)

4. VPP Uncertainty Management

Minimizing the variability of renewables into the grid network, especially WP, entails detailed modeling of energy generation assets. By proposing a probabilistic modeling approach, both system parameters and variables are introduced and allocated a threshold that could secure the grid network and minimize extreme power supply and demand deviations. Equations (17)–(20) are as follows:

$$B_t=b_t-\left(P^{vpp}+U_t\right), \forall t\in T$$

(17)

$${\mathbb{R}}_t\left[B_t\le {\delta }_t\right]\ge 1-{\epsilon }_t \forall t\in T$$

(18)

$$v\left({\mathrm{P}}_t\right)={\int }_{b_t\in T}\left(\right[\mathcal{l}n{\mathbb{P}}_t-{\mathcal{l}n\mathbb{R}}_t\left]{\mathbb{P}}_t\right){db}_t\forall t\in T$$

(19)

$${\mathbb{Z}}_t=\left\{{\mathbb{P}}_t\left|v_t\right.\left({\mathbb{P}}_t\right)\le V_t\right\}, \forall t\in T$$

(20)

where $b_{t}is$ used to represent real-time energy shortfall under the expected true probability ${\mathbb{P}}_t$; since it cannot be quantified at the first stage of the optimization process, a reference distribution ${\mathbb{R}}_t$ is applied, where $b_t$ [33] historical samples are usually used to obtain a reference distribution. For the two distributions to be applied, they have to be symmetrical about the mean. In the literature, techniques, such as Wasserstein distance and the Kullback–Leibler (KL) method have been used in [34,35]. The KL method [33] has been used to quantify the divergence between the two distributions as defined by Equation (19), while ${\mathbb{Z}}_t$ represents ambiguity set in Equation (20). Equation (21) is as follows:

$$\underset{{\mathbb{P}}_t\in {\mathbb{Z}}_t}{inf}{\mathbb{P}}_t\left[B_t\le {\delta }_t\right]\ge 1-{\epsilon }_t, \forall t\in T$$

(21)

$B_t$ is further compared to ${\delta }_t$ which is a given constant that has a probability interval of $1-{\epsilon }_t$ for all $t$. In order to obtain Equation (23), Equation (17) is input into Equation (21).

$${\mathbb{R}}_t\left[b_t\le \left(P^{vpp}+U_t+{\delta }_t\right)\right]\ge 1-{\epsilon }_t \forall t\in T$$

(22)

Finally, Equation (17) is approximated by Equation (24).

$${\mathbb{R}}_t\left[b_t\le {b^{11}}_t\right]\ge 1-{\epsilon }_t \forall t\in T$$

(23)

$$X\cong P^{vpp}+U_t+{\delta }_t\ge b_t^{11} \forall t\in T$$

(24)

Two objective functions are given by both Equations (25) and (37). The rest of the equations are balancing equation constraints (Equation (26)) and constraints that define the operation of the VPP endogenous DERs.

$$\underset{P^{vpp},P^{BD},P^G,P^{EC},P^{ED}}{Min}\sum _{t=1}^T\sum _{i^G\in \Omega G}^G{C}_{,t}^G+\underset{i^B\in \mathsf{\Omega }\mathrm{B}\mathrm{E}\mathrm{S}\mathrm{S}}{\overset{BESS}{\sum C_{i,t}^{BESS}}}+{\xi }^{op}{P}_t^{VPP}$$

(25)

$$P_t^{VPP}=\sum _{i,g}{P}_{it}^G+\sum _{i,BESS}{P}_{it}^{BESS}+\sum _{i,w}{P}_{it}^W+\sum _{i,pv}{P}_{i,pv}^{pv}+P_{it}^{BD}$$

(26)

$$V_{\min }\le V_j\le V_{\max }$$

(27)

$$V_{-i}\le V_j\le {\bar{V}}_i$$

(28)

$$P_{min}^{vpp}\le P_{t,s}^{vpp}\le P_{max}^{vpp} ,\forall t$$

(29)

$$P_{min}^{BD}\le P_{t,s}^{BD}\le P_{max}^{BD} ,\forall t$$

(30)

$$B_t=b_t-\left(P_{it}^{vpp}+P_{it}^G\right),\forall t\in T$$

(31)

$$\underset{{\mathbb{P}}_{t\in \mathbb{Z}}}{ inf }{\mathbb{P}}_t \left[B_t\le {\delta }_t\right]\ge 1-{\epsilon }_t, \forall t\in T$$

(32)

$$\sum _{i,g}{P}_{g,t+1}^G-\underset{i,g}{\sum P_{g,t}^G\le R_g^{GU},\forall t<T,\forall g}$$

(33)

$$\sum _{i,g}{P}_{g,t}^G-\underset{i,g}{\sum P_{g,t+1}^G\le R_g^{GD},\forall t<T,\forall g}$$

(34)

$$\sum _{t=1}^{MUP}{\alpha }_{g,t+1}^G-1\ge MU{P}_g^G,\forall {\beta }_{\mathit{g},\mathit{t}}^{\mathit{G}}=1$$

(35)

$$\underset{t=1}{\sum ^{MUP}}1-{\alpha }_{g,t+1}^G\ge MD{N}_g^G,\forall {\mathit{\gamma }}_{\mathbf{g},\mathbf{t}}^{\mathbf{G}\mathbf{B}}=1$$

(36)

$$\begin{array}{c}\underset{{\mathit{P}}^{\mathbf{V}\mathbf{P}\mathbf{P}},{\mathit{P}}^{\mathbf{D}\mathbf{B}},{\mathit{P}}^{\mathbf{G}},{\mathit{\xi }}^{\mathbf{C}}}{Max Profit}\\ ={\displaystyle \sum _{t=1}^T}{\xi }^{\mathbf{C}}{\mathbf{P}}_{\mathbf{t}}^{\mathbf{V}\mathbf{P}\mathbf{P}}-\sum _{t=1}^T{\chi }_t^{inC}{\mathit{P}}_{\mathbf{t}}^{\mathbf{D}\mathbf{B}}\Delta t-\underset{i^g\in \Omega G}{\stackrel{G}{\sum {\gamma }_t^g}}{\mathit{P}}_{\mathbf{i},\mathbf{t}}^{\mathbf{G}}+ \end{array}\underset{i^g\in \Omega G}{\sum ^G}SU{C}_g^{VPP}{\mathit{\beta }}_{\mathbf{g},\mathbf{t}}^{\mathbf{V}\mathbf{P}\mathbf{P}}$$

(37)

4.1. DR-JCCO-Approximating Algorithms

Convex approximations can be very handy in tracking DROPs. CVaR is known to be the best approximation of DROPs as it enables tractability of DROPs and reduces the conservatism of results; however, it can vary from case to case. The following subsection introduces mathematical notations to illustrate how the two concepts differ within the context of energy systems. The H-X algorithm is introduced as a new convex approximation algorithm; according to the study in [36], it is the best so far in approximating DROPs.

4.1.1. CVaR Application

In the literature, VaR is typically used to estimate investment risk by setting a predefined threshold level. However, CVaR extends beyond VaR by capturing the expected shortfall that occurs beyond this threshold. The mathematical formulations for VaR and CVaR are presented in Equation (38) and Equation (39), respectively.

Let $\tilde{X}$ be a random variable and denote its probability distribution and CDF by $\mathbb{P}$ and $Q_{\tilde{\mathit{X}}}(\cdot )$, respectively. At a given risk level $\epsilon ,\left(1-\epsilon \right)$, the VaR of $\tilde{X}$ can be represented by the following Equation (38):

$${VaR}_{1-\epsilon }\left(\tilde{\mathit{X}}\right):=\underset{s}{min}\left\{s:Q_{\tilde{\mathit{X}}}\left(s\right)\ge 1-\epsilon \right\}$$

(38)

$${CVaR}_{1-\epsilon }\left(\tilde{\mathit{X}}\right):=\underset{\beta }{min}\left\{\beta +{\displaystyle \frac{1}{\epsilon }}{\mathbb{E}}_{\mathbb{P}}[\tilde{\mathit{X}}-\beta {]}_{+}\right\}$$

(39)

4.1.2. H-X

The modeling approach for the H-X algorithm is structurally similar to that of CVaR. However, the key distinction lies in the iterative procedure, where the algorithm searches for the optimal value of $t$ by evaluating between predefined lower and upper bounds. The minimum value obtained within this range is then assigned to $t$, thereby enhancing the tractability and feasibility of the problem, as shown in the following Equation (40).

$$X^{\ast }(t):=\underset{x\in X,s(\cdot )\ge 0}{\mathit{min}t}\left\{E\left[s\left(\tilde{\xi }\right)\right]:\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \left(4.40\right),c^{\top }x\le t\right\}$$

(40)

4.1.3. General MPEC Formulation

The MPEC approach facilitates the transformation of a two-objective function into a single-objective formulation, enabling faster solutions through the use of available optimization solvers. Equations (25) and (37) are further modeled using a binary expansion technique to derive a single-level objective function that incorporates all relevant constraints. When implementing the MPEC method, both complementarity constraints and the Big-M notation are carefully considered.

In this model, the VPP output and market prices are treated as decision variables, as defined in Equation (A7). Their interaction results in a nonlinear expression, which poses challenges for standard solvers. To ensure tractability, the problem must be reformulated as a mixed-integer linear programming (MILP) model using appropriate mathematical notation. The linearization process is detailed in Appendix A, where the Big-M parameter $B^M$ must be sufficiently large to ensure an accurate transformation without compromising model integrity. The bilinear terms involved are represented as $x=P^{VPP}$ and ${y=\xi }^{op}$.

5. Results

The modeling process utilizes the parameters and decision variables defined earlier, with the optimization problem solved using the Gurobi solver. The simulations were conducted on a system with the following specifications: Intel(R) Core(TM) i5-6200U CPU, with 2.40 GHz and 16.0 GB RAM (15.9 GB usable). As outlined in Table 1, the proposed bilevel framework facilitates the design of a VPP capable of participating in the electricity market and evaluating the profit margins for VPP operators. This dual-objective structure enables the VPP to coordinate the internal scheduling of generation units by minimizing operational costs, while supporting participation in the DM, which operates on a 24 h basis.

To address the optimization problems, different structural formats were adopted. The CVaR formulation was implemented using CVXPY, while the H-X algorithm, due to its iterative nature in determining the optimal value t within upper and lower bounds, was structured and executed using Python 3.13.2 and spyder 5.5.1 as the integrated development environment (IDE) structured in the Gurobi 11.0.0 format. WP scenarios were generated and integrated into the optimization model to reflect variability in wind generation. These scenarios serve to capture the extent of wind power fluctuations. Figure 2 illustrates the set of generated wind power scenarios.

The optimization model incorporates uncertainty bands that represent deviations in power generation, with both the maximum and minimum power levels illustrated in Figure 3. These quantified deviations define an uncertainty region, which allows the decisionmaker to evaluate the influence of such uncertainties on the overall performance of the power system network.

Figure 4 illustrates the generated power along with the corresponding region used to evaluate the level of uncertainty associated with the wind power plant.

5.1. DR-JCCO-CVaR

The CVaR method applied to ensure the tractability of the DR-JCCO model yields results that demonstrate how a bilevel VPP structure can effectively coordinate the dispatch of DGs to provide baseload, alongside the BESS and WPP. The model incorporates uncertainty scenarios, and the resulting variability allows the VPP to adapt its output in coordination with DERs to meet prevailing demand.

Figure 5 illustrates a scenario of BESS underutilization, which arises due to DG1 and DG2 being dispatched at their maximum capacities instead of prioritizing the BESS. In this case, the maximum charging and minimum discharging rates for the BESS are set at 100 MW. The figure offers a detailed depiction of power contributions from the VPP and DGs. At the boundaries of the time horizon, the VPP matches the demand profile by dispatching the required power. Around hour 7, the power supply stabilizes until approximately hour 12, after which the VPP output begins to decline, as indicated by a reduction in DG1 and DG2 contributions. Furthermore, the figure highlights the underutilization of the BESS, which is primarily attributed to the prioritization of the power supply from the DGs.

Figure 6 presents the expanded profile of the BESS storage level over time. Proper sizing of the storage system is critical and must align with the appropriate specifications and operational requirements, such as battery capacity, maximum and minimum state of charge (SoC), C-rate, and depth of discharge (DoD). With accurate sizing, BESS monitoring becomes more manageable, ensuring an optimal charge–discharge cycle. The BESS capacity essentially defines the amount of energy that can be supplied to meet the prevailing demand.

In this scenario, Figure 5 reflects the underutilization of the BESS, as only a limited amount of power is discharged. Discharging occurs briefly around hours 2 and 4, followed by a gradual increase in charging power, which stabilizes between hours 6 and 16 before reaching approximately 135 MWh. The marked minimum and maximum SoC thresholds remain unaffected due to the BESS being underutilized in this instance. The total operational cost under the CVaR-based approach amounts to USD 490,906.39, indicating an increase compared to when DR-JCCO is applied with H-X, as will be seen in subsequent sections.

The second demonstration case is under the CVaR framework involving the scheduling of DG1 and DG2, where DG1 operates at a constant output of 300 MW and DG2 at 400 MW continuously over the 24 h period. The BESS is then dispatched to meet the remaining portion of the demand, as illustrated in Figure 7 and Figure 8, respectively. The dispatch decisions for the generators are influenced by their respective fuel cost parameters, whereby DG1 is prioritized due to its lower operational cost relative to DG2.

An expanded illustration of the BESS storage level under SoC constraints is presented using a smoothed SoC curve, as shown by Figure 8. The initial SoC is at 50% and, after undergoing charging and discharging cycles, the BESS retains the same energy level. It is important to note that the storage capacity is limited by predefined SoC bounds, with the maximum and minimum levels set at 90% and 20%, respectively. Additionally, the smoothen curve in Figure 8 is not always representative of actual physical systems, where the profile may vary depending on operational conditions. The curve can be compared to the one in Figure 9.

Meanwhile, Figure 10 presents the power output from DG1, which operates at a constant maximum capacity of 300 MW and incurs the lowest fuel cost. DG2, on the other hand, supplies up to 400 MW and is scheduled to adjust its output based on the prevailing system demand. Alongside the BESS, these resources are coordinated to meet the total energy requirement. The VPP aggregates and dispatches the units optimally, closely following the illustrated demand profile. The peaks observed in the VPP output curve highlight the method’s responsiveness in aligning with demand fluctuations. The BESS discharges primarily at the two ends of the time horizon, as depicted in the charge–discharge profile in Figure 10. Furthermore, the CVaR-based algorithm introduces slight variations in the dispatch results, such as the charge–discharge behavior shown in the same figure. According to the figure, the BESS initiates charging between hour 5 and hour 10 with relatively high active power; this is followed by a decrease in charging rate and a brief spike at approximately hour 13. Subsequently, the charging power becomes zero. From hour 15, the BESS remains idle, neither charging nor discharging, until hour 21, when the BESS starts discharging. The total operational cost in this case is approximately USD 342,800.23, indicating that the BESS is both appropriately sized and efficiently scheduled.

Curtailment planning was not included in the model, as shown in Figure 11, since the generated WP was considered sufficient. Consequently, the model concentrated on dispatching DGs and the BESS. The H-X algorithm performed curtailment of the wind power to demonstrate advanced modeling of DERs under continuous scheduling.

In summary, the total profit achieved using the CVaR approach was approximately USD 500,000, aggregated over the scenarios generated. This outcome, depicted in Figure 12, is based on the assumption that the traded quantities correspond closely to the utilized scenarios and the observed power levels.

5.2. DR-JCCO-H-X

The H-X algorithm employs an iterative process to monitor the DR-JCCO and its convergence, effectively reducing the conservativeness typically associated with DROPs, as demonstrated in this section. Once the optimal value of t is determined, all VPP resources are scheduled to satisfy the demand. Figure 13 illustrates the scenarios generated relative to the optimal t value.

When applying the H-X algorithm to discharge the BESS, the SoC decreases very quickly, indicating that the iterative loop converges to optimality rapidly. Due to the high time efficiency of H-X, the BESS is dispatched to meet demand whenever the supply from DG1 and DG2 is insufficient. Around hour 20, the SoC reaches zero, as shown in Figure 14. The spikes in these figures represent the BESS’s charge–discharge cycles, which could raise concerns regarding the battery’s long-term health and safety. Although this method is fast and accurately approximates the DR-JCCO problem, frequent charge–discharge cycles may reduce the BESS’s efficiency over time.

Additionally, Figure 15 illustrates wind power generation, curtailment, and an adjusted demand profile. Curtailment is implemented by applying a percentage of wind power for reduction along with penalization parameters. In cases of wind power surplus, 80% of the generated wind power is utilized, while the remainder is curtailed using the wind usage fraction (WUF) and curtailment fraction (CF). For demonstration, the model uses values of 0.9 for WUF and 0.5 for CF.

The instantaneous response of the H-X algorithm in dispatching distributed generation assets makes it well-suited for ensuring continuous power supply within power networks. Furthermore, the demand curve in Figure 15 has been adjusted to reflect realistic, time-varying demand conditions.

This strategy is crucial for evaluating the effects of overgeneration in the grid system. For example, Figure 16 shows wind curtailment peaks at approximately 0.12 MW when a WUF of 0.9 is applied, which is relatively low compared to a curtailment fraction of 0.5, whereby curtailed wind power is about 2.9 MW at maximum, as indicated in Figure 15.

The figures can be rescaled to provide a clearer representation of the BESS charge–discharge cycles alongside the power output from DG1 and DG2. Specifically, Figure 17 offers a zoomed-in view of scenarios where dispatchable generators are activated, particularly during periods when the VPP’s total supply falls short of demand. The figure illustrates the underutilization of the BESS, marked by brief and rapid charging and discharging periods. This limited utilization is attributed to the adequacy of the overall VPP supply in meeting the system’s demand, thereby relegating the BESS to cover only minor fluctuations. Furthermore, the BESS operational constraints are enforced, with the minimum SoC set at 20% and the maximum SoC of 90% indicated by a red dashed line. DG2 and DG1 operate during high-demand hours, approximately at hours 22 and 23, respectively. This dispatch strategy is informed by the algorithm’s capability to account for operational cost factors, including fuel usage, as well as startup and shutdown costs. Consequently, the method supports the efficient coordination and deployment of DERs through the VPP framework to reliably meet the demand.

Subsequently, the VPP schedules the dispatch of its generation assets by first coordinating internal resources through the DR-JCCO-H-X framework. As shown in Figure 18, this coordination mechanism enables the VPP to dynamically respond to demand fluctuations, with the energy output from the BESS decreasing progressively over time. The approach also safeguards the operational integrity of the BESS by ensuring that its charge–discharge cycles remain within the defined state-of-charge limits.

Furthermore, the interdependence between the BESS storage capacity and the generation capacity of DGs has been established. For example, if the BESS has a storage capacity of 400 MWh while the DGs are inadequately sized (e.g., 50 MW and 100 MW), the BESS tends to remain underutilized. This underutilization translates to reduced VPP profitability, as infrequent charge–discharge cycles are scheduled to avoid high operational expenses, particularly those associated with dispatching the DGs solely for BESS charging, which can be prohibitively costly. To avoid escalating costs, the VPP preferentially dispatches other internal resources to meet the demand while minimally charging the BESS. Consequently, the generation capacity of DG1 and DG2 directly influences the charging and discharging behavior of the BESS, as illustrated in Figure 18. Notably, the figure shows a decline in BESS activity, with a complete discharge occurring around hour 19, after which the VPP relies solely on DGs to satisfy the demand. It is important to note, however, that in practical applications, frequent and rapid charge–discharge cycles are generally discouraged, as they may compromise the structural integrity of the storage cells and accelerate the degradation of the BESS over time.

Moreover, as illustrated in Figure 18, the limited capacity of the BESS enabled the VPP to optimally schedule the majority of its available generation resources. This is further supported by Figure 19, which displays the profits that can be obtained with respect to the specific size of the BESS.

Furthermore, Figure 20 and Figure 21 illustrate the variation in the demand profile when wind power is integrated into the VPP’s energy portfolio. The overall demand notably decreases due to the contribution of wind generation. Despite this, the curtailment of wind power remains necessary to effectively manage surplus renewable energy and maintain supply–demand balance. In this scenario, the curtailed portion of wind power is strategically utilized to charge the BESS, thereby enhancing system flexibility and storage efficiency.

Meanwhile, Figure 22 presents the calculated profits resulting from the VPP’s operation using the H-X algorithm. These profits are derived based on the system demand and the VPP’s capability along with its internal resources to supply sufficient power to meet that demand.

Regarding the use of the binary expansion (BE) method, a modeling approach is adopted to develop a bidding strategy for the VPP within the proposed bilevel optimization framework. This model is specifically designed to linearize continuous variables, which are otherwise difficult to handle using standard solvers, like Gurobi or CPLEX. The mathematical formulation defines system-level constraints for storage units, distributed generators, and the VPP itself. As a result, the VPP is able to efficiently dispatch DGs and the BESS in response to demand while adhering to a predefined threshold.

The binary expansion technique reformulates the VPP model by addressing continuous variables whose multiplication leads to nonlinear expressions that are challenging to solve. The profit maximization objective function (Equation (37)) is simulated under the imposed system constraints outlined in Appendix A. Initially, the approach facilitates the cost-effective dispatch of DG1 and DG2 generators. For example, when DG1 has a fuel cost of USD 2.58/MMBtu, the algorithm minimizes operational costs by prioritizing the generator with the lowest cost. As depicted in Figure 23, the VPP dispatch is constrained to a maximum of 2000 MW, a limit imposed by the TSO. Within this boundary, the VPP coordinates its internal resources to ensure real-time demand fulfillment.

The dispatch schedule for DG1 and DG2, as illustrated in Figure 24, is determined based on their respective fuel consumption costs, as mentioned earlier. Additionally, the BESS is utilized to supply power between hour 5 and hour 13, thereby enabling the VPP to fulfill the system demand during this period.

As the total generation capacity of the VPP increases, the BESS is dispatched more frequently to support load demand. However, the VPP’s output remains constrained by the capabilities of its internal assets. For example, when a 3000 MW load is required, the VPP can only supply up to 2800 MW. Nonetheless, the complete dispatch of all available generators remains a key priority for the TSO to ensure demand is met. Figure 25 illustrates the 24 h dispatch schedule of VPP output, with power oversupply observed at hour 2 and hour 18.

Figure 26 illustrates a case where a 200 MWh BESS is charged and discharged at a fixed rate of 100 MW, applying the BE method as a standalone approach. In this instance, the storage profile follows a largely linear trend, steadily increasing and then slightly decreasing during discharge events. In contrast, the CVaR and H-X formulations incorporate complementarity constraints that are specifically designed to manage nonlinearities present in the system model, rendering them more suitable for long-term BESS operational strategies.

This study has evaluated the performance of three principal optimization techniques. In addition to assessing operational cost minimization and profit generation under uncertainty, the methods were analyzed using several key performance indicators, including the number of nodes explored, simplex iterations, computational time (in seconds), objective value, number of solutions identified, and the optimality gap metric, which reflects the quality of the best solution relative to the best known bound. These indicators collectively facilitate the evaluation of model performance in terms of solution accuracy, computational efficiency, and algorithm robustness.

Figure 27 provides a comparative analysis; the results reveal that the DR-JCCO approach incurs the highest number of simplex iterations, followed closely by the DR-JCCO-H-X method. This observation supports the notion that increased iteration counts correspond with a greater number of explored nodes and improved solution quality, as evidenced by the performance of the CVaR-based formulation.

Additionally, Figure 28 presents a comparison of the objective values achieved by each method, which correlates with the number of solutions identified. On average, the solution count is lower for the DR-JCCO method compared to the H-X and CVaR algorithms. The higher objective values observed for the H-X and CVaR approaches indicate a greater number of feasible solutions or a faster convergence rate, thereby demonstrating superior overall performance.

The results presented in

Table 2. Performance assessment of optimization techniques.

Optimization Approach	Performance Metrics
	Sum of Iteration	Summing Nodes Explored	Sum of Time (s)	Sum of Gap	Summing Simplex Iterations	Conservativeness of Solutions	Tractability Level
DR-JCCO	15	17	2.73848223	0.022970004	85	High	Low
DR-JCCO-CVaR	15	27	5.54975679	0.028598226	157	Moderate	High
DR-JCCO-H-X	15	16	5.90154595	0.021566176	137	Low	Best

reveal that the H-X algorithm achieves a better solution gap than CVaR, although it yields lower solution quality and requires more time to find a feasible solution. In contrast, the DR-JCCO method requires less computational time compared to the other two approaches, with fewer iterations, which aligns with its higher solution conservativeness. Table 2 also compares the methods in terms of solution conservativeness and tractability: CVaR and H-X algorithms reduce solution regularity and enhance the tractability of general DR-JCCO problems. whereas DR-JCCO alone exhibits higher solution regularity but very low tractability despite its faster convergence, as illustrated in Figure 27.

Figure 29 demonstrates that fewer iterations or explored nodes do not always lead to better solution quality; rather, they may signify lower computational effort, which could come at the cost of solution accuracy.

6. Discussion

The results show the ability of the VPP to meet most of the demand, with residual load handled by the BESS in most Cases. Note that under uncertainty involving the demand variation attributed to both supply and energy price volatility. The demand could fluctuate with increase or decrease in energy prices, and this is where the optimization techniques play a crucial role by limiting extreme variations that can cause peak demand in the early mornings and evenings. To monitor and manage the demand in a more optimal manner, the BESS steps in to alleviate peaks and allocate energy supply to regions which are insufficiently fed by the available energy generation resources to meet the available demand. The DR-JCCO in its pure form effectively manages uncertainties related to renewable energy sources and conventional generation assets within predefined thresholds; however, it is limited, as it is less tractable and suffers from a regularity of solutions leading to compliance challenges during the optimization process of the BESS and DGs.

The simulations conducted using the CVaR and H-X algorithms present a distinctive approach to solving JCC problems. Usually, the demand is not fully satisfied with DR-JCCO, suggesting possible supply shortages. Despite this, DR-JCCO remains a leading approach that integrates distributional and robust optimization to effectively schedule and operate DERs.

It is imperative to note that choosing the most suitable method involves a trade-off and largely depends on the specific problem context. These methods demonstrate enhanced performance when integrating BESS and renewables into the grid. Although CVaR delivers high-quality results, the H-X algorithm emerges as a strong alternative for sizing the BESS due to its faster convergence, reduced computational complexity, and improved problem tractability. The H-X method results in rapid BESS charge–discharge cycles while maintaining efficient overall handling of the DR-JCCO problem. Notably, H-X benefits from determining an optimal parameter $\mathrm{t}$, which is achieved after nearly all scenarios have been evaluated and the probability threshold remains feasible, as illustrated in Figure 13.

It is essential to carefully select constraints to optimize the system’s use within safety limits. Minimizing charge–discharge cycles helps extend the lifespan of the BESS, although durability is also influenced by other factors, such as cycling aging, depth of discharge (DoD), charge rates, and thermal conditions. Figure 5 illustrates how battery stress is mitigated by reducing the cycling frequency while increasing the power supply from DGs. However, even if the BESS remains idle, its longevity is not guaranteed due to calendric aging effects [37], which degrade cells chemically over time. Therefore, minimizing idle periods is recommended to slow self-degradation.

Overall, the CVaR framework promotes smooth BESS operation and optimally schedules DGs to meet demand efficiently. Given DG1’s startup and heat rate costs of USD 10.5/MMBtu and DG2’s at USD 11/MMBtu, with respect to the startup costs of USD 100 and USD 120 per event, the optimization prioritizes DG1 by maintaining a steady output (e.g., 300 MW over 24 h). When demand exceeds DG supply, the BESS and VPP coordination balances the system. In instances of BESS underutilization, DGs fulfill most of the demand. The total operational cost with this approach was approximately USD 490,906.

Applying the H-X algorithm allows easier tracking of profits and reduces operational costs by limiting DG use; the more expensive DG is reserved for peak hours to cover residual load. Our results confirm that effective VPP scheduling depends on the appropriate sizing of both the BESS and DGs, as an oversized BESS with low wind and DG production leads to underutilization of the BESS. Power generated by DGs can be stored for peak consumption, and curtailed wind power can supplement the energy mix. Also, sizing decisions must consider both demand and DG power output. Typically, DGs operate continuously over 24 h to avoid cold start costs, with the VPP regulating generation during low demand to charge the BESS for later dispatch during peak demand. Hence, the H-X algorithm provides numerous performance advantages, enabling BESS discharge based on the optimal parameter $\mathrm{t}$ and scenario sets. To explore changes in BESS storage capacity, further robust analysis through parameter tuning is necessary. The CVaR results closely aligned with H-X in terms of BESS charge–discharge behavior. Differences in VPP costs and profits among the two methods arise partly from model parameters and constraints embedded in the optimization formulations.

Finally, it is important to note that the CVaR and H-X strategies schedule generation units and coordinate resources to follow the demand profile, therefore placing BESS at the core of the optimization.

6.1. Conclusions

The findings highlight the successful application of advanced optimization techniques for integrating renewable energy sources into the power grid. This article’s objective was established through a thorough review of the existing literature, supporting the adoption of DR-JCCO due to its advantages of reduced computational burden and lower information loss compared to traditional methods, such as RO and SP. Achieving the right balance in integrating BESS, DGs, and RERs allows for a comprehensive assessment of grid reliability and the quantification of uncertainties’ impact on the overall safety of power transmission networks. It was noted that conventional approaches, like RO and SO, are hindered by significant information loss and high computational demands. These drawbacks motivated the implementation of a DR-JCCO framework within a bilevel optimization structure for modeling the VPP. In terms of improving tractability, existing approaches, such as CVaR, although popular, were found to be inadequate to fully meet the study’s objectives. This research advances the field by proposing a bilevel, data-driven model tailored to energy systems, which integrates distributional robustness with resilience considerations while minimizing information loss. Emphasis was placed on implementing algorithms capable of effectively solving the problem, notably the H-X algorithm, a novel contribution to integrated energy system modeling. The demonstrated effectiveness of the H-X algorithm in reducing operational costs and optimizing BESS utilization underscores its potential superiority and suitability within the DR-JCCO framework.

6.2. Future Work

The VPP model incorporating WP and DGs presents a novel approach to integrating RESs into the grid network. The overall mathematical modeling of the VPP in a bilevel framework was explored as seen in Appendix A, with BE being employed to linearize nonlinear models, laying the groundwork for further investigation into alternative mathematical tools that could enhance the linearization process.

Additionally, the following sub-topics will need to be considered in subsequent studies:

• Improvement of the H-X algorithm: Further research is needed to enhance the computation of the optimal t values within the H-X algorithm. This would improve its ability to regulate the charge–discharge cycles of the BESS, thereby facilitating more efficient long-term usage of the storage system and reducing degradation rates during operation.
• Dispatch of multiple VPPs: A potential area for future exploration involves the inclusion of multiple virtual power plants (VPPs). This would enable advanced analysis of coordinated scheduling and dispatch mechanisms across interconnected VPPs, a critical consideration for accurate and equitable settlement procedures in electricity markets.
• Advancement of EPEC models: While MPECs have received considerable attention, EPECs remain underexplored, specifically in the context of distribution networks. Future work should aim to bridge this gap.