Placement and sizing of photovoltaic and bio-waste unit with hydrogen storage considering economic energy management of intelligent distribution network

Abstract

A virtual power plant integrates diverse energy sources and storage systems, functioning as producers within a network. Strategically determining its optimal placement and the size of its components leads to notable improvements in key economic and technical metrics of the network. This research builds a framework for sizing and positioning renewable virtual power plants, incorporating hydrogen storage systems as part of a broader multi-objective energy management strategy for smart grids. The proposed approach operates as a bi-level optimization model. In the upper-level model, it addresses technical, financial, and environmental objectives from the perspective of the distribution system operator. The aim is to minimize the total weighted operating costs, energy losses, and emissions within the distribution network. The model considers constraints such as the AC power flow equations for smart grids, alongside operational and voltage security restrictions. Meanwhile, the lower-level model focuses on the placement and sizing of renewable virtual power plants incorporating hydrogen storage systems. Here, the goal is to reduce planning costs, with constraints tied to the operation and flexibility models of renewable sources and storage systems. To integrate these models effectively, the Karush–Kuhn–Tucker (KKT) conditions are applied to create a unified single-level framework, while a Fuzzy decision-making technique facilitates the derivation of compromise solutions. This approach also accounts for uncertainties related to load demand, renewable energy outputs, and fluctuating energy prices using scenario-based stochastic optimization combined with the Kantorovich method and Roulette Wheel Mechanism. For solution optimization, Red Panda Optimization (RPO) is employed, demonstrating superior convergence speed and precision in comparison to other solvers. The research incorporates innovative elements by addressing optimal placement and sizing of flexible virtual power plants within active distribution grids, while factoring in bio-waste energy management and the role of hydrogen storage capacities in network operation and security. Evaluation through numerical case studies highlights significant enhancements in grid performance. In comparison to conventional load flow methods, the proposed solution optimizes operations by reducing network operating costs, improving voltage security, and mitigating environmental impacts. When adopting a compromise solution framework, the distances of network operation costs, energy losses, and emissions from their minimal achievable values are by 14%, 29%, and 21%, respectively. Moreover, RPO demonstrates remarkable precision with a final solution’s standard deviation of approximately 0.97%, effectively identifying optimal results with enhanced convergence efficiency. This study underscores the transformative potential of virtual power plants in improving energy management and distribution grid planning.

Introduction

Motivation

The global transition toward renewable energy sources (RESs) are rapidly advancing, spurred by growing concerns over the environmental consequences of unchecked fossil fuel consumption. Among the myriad solutions, solar panels, commonly known as photovoltaic systems (PVs), have emerged as a popular option due to their relatively compact installation requirements and reduced labor needs compared to other renewable alternatives¹. Nevertheless, deploying a greater number of PVs within the distribution network introduces operational complexities, particularly when their energy output is not properly regulated. For example, PVs can contribute to overvoltage issues within the network. In addition to solar technology, bio-waste units (BUs) also play an integral role in clean energy generation by converting environmental waste into electricity. However, the power output from both PVs and BUs is inherently variable and can result in discrepancies between projected energy production during the day-ahead forecasting phase and actual real-time performance². This variability often leads to imbalances between electricity generation and consumption during active network operation. To mitigate these challenges, integrating storage systems alongside PVs and BUs provides a crucial stabilizing mechanism, ensuring consistent network performance and reducing fluctuations between forecasted and actual output². Effective management of energy resources within the electrical distribution network is paramount for realizing these goals. A critical strategy involves harmonizing PVs, BUs, and storage systems into an interconnected framework that works in tandem with the distribution system operator (DSO). The DSO serves a pivotal role in optimizing technical, environmental, and economic parameters across the network. To unlock the full potential of these energy resources, the concept of a virtual power plant (VPP) can be employed, essentially transforming PVs, BUs, and storage units into a unified entity designed to enhance operational efficiency and sustainability within the distribution network. However, the success of such an initiative hinges on identifying the optimal size and location for VPPs. Inadequate positioning or scaling could drastically undermine their effectiveness, highlighting the importance of striking the right balance. Ultimately, precise implementation is key to achieving robust and sustainable energy management outcomes for modern electrical distribution systems.

VPP can operate in grid-connected mode, where maintaining the technical and economic stability of the network becomes paramount. To achieve this stability, the energy management system within the VPP must seamlessly align with the financial and operational objectives of the distribution network. Effective bidirectional coordination between the DSO and the VPP operator is essential to ensure smooth operation. The placement of a VPP within the network emerges as a critical factor, as it has a significant impact on various performance outcomes. Poor location choices can introduce challenges that negatively affect network stability, rendering even sophisticated energy management strategies less effective in mitigating these issues. This underscores the importance of selecting an optimal location for the VPP to ensure efficient operation and minimize potential disruptions. Equally important is the optimization of resource capacity and storage systems within the VPP. Scaling these components improperly could compromise energy management performance and hinder the full utilization of their capabilities. For example, an imbalance where resource capacity is inadequate but storage systems are oversized might result in underutilized storage due to limitations in charging and discharging power. Such scenarios reduce operational efficiency and prevent the VPP from achieving peak performance. To ensure maximum effectiveness, a dual focus is necessary: strategically positioning the VPP within the network and optimally scaling its resources and storage systems. By addressing both aspects, the performance, reliability, and overall efficiency of the VPP can be significantly enhanced, aligning its capabilities with the dynamic needs of the distribution network.

Literature review

The domain of energy management in distribution networks has experienced significant advancements in recent years. The study referenced in³ introduces the concept of economic flexible-securable operation (EFSO), specifically designed for smart distribution networks (SDNs). This methodology prioritizes the integration of distributed generation and storage technologies to support eco-friendly energy production. Framed as an optimization paradigm, EFSO aims to minimize operational costs while adhering to core constraints such as optimal power flow, environmental sustainability, security, and flexibility. Expanding on these developments, the research cited in⁴ presents a novel strategy for optimizing grid-connected micro-grid operations. This approach takes into account the uncertainties linked to renewable energy sources within distributed generation (DG) units, such as wind turbines and photovoltaic systems. By addressing these challenges, the proposed strategy significantly boosts efficiency in energy distribution across dynamic and unpredictable scenarios. Continuing this trajectory, the work published in⁵ delves into expected energy not supplied (EENS) alongside the voltage stability index (VSI) within dynamic reconfiguration frameworks. This research explores both balanced and unbalanced distribution networks while integrating RESs and energy storage systems (ESSs). Its insights contribute to the development of more resilient and adaptable network infrastructures capable of coping with ever-changing grid requirements. To address challenges in electric vehicle (EV) charging infrastructure, the study outlined in⁶ introduces a sophisticated management model for EVs connected to grids powered by wind and solar energy. Classifying EVs into four distinct types based on their interaction patterns with grid operations, the model employs a normal distribution function to account for their stochastic dispersion across classes. Moreover, it effectively incorporates the fluctuating nature of wind speeds and solar irradiation, offering solutions to mitigate coordination complexities presented by these RESs.

A diverse range of innovative approaches has been explored across recent studies to address intricate challenges within energy management and smart distribution networks. Study⁷ introduces a Multi-Agent Deep Reinforcement Learning (MADRL) control framework using an enhanced actor-critic algorithm with a shared attention mechanism. This framework is tailored for efficient energy management in greenhouse networks. Research in⁸ examines the dynamic integration of price-responsive demands, inverter-based photovoltaic units, and battery energy storage systems (BESSs) within active distribution networks (ADNs), enabling active participation in real-time energy markets. Expanding beyond foundational strategies, Study⁹ focuses on optimizing energy management for microgrids, incorporating battery charging/swapping stations alongside renewable energy sources. Reference¹⁰ adopts a multi-objective Tuna Swarm Optimization technique for ADN applications, leveraging advancements in multi-objective transformation, initialization processes, and population diversity to enhance results. Similarly, Reference¹¹ introduces the Golden Jackal Optimization (GJO) method to efficiently address energy management complexities in distributed-generation systems equipped with hybrid sources and BESSs. In disaster scenarios, Reference¹² proposes a two-objective optimization method aimed at minimizing both energy deficits caused by natural disasters and annual planning costs for virtual power plants (VPPs). Reference¹³, on the other hand, develops a three-stage stochastic bi-level optimization model to maximize VPP profits through upper-level strategies, while the lower levels focus on distributed energy resource decision-making processes for VPP selection. Collaboration among heterogeneous microgrids forming a VPP is explored in Reference¹⁴, where system synergies are emphasized to boost cooperative efficiency. Reference¹⁵ studies the optimal placement and sizing of flexible renewable VPPs in smart distribution networks using a bi-level optimization framework. Lastly, Reference¹⁶ devises a comprehensive three-step voltage regulation coordination scheme. This approach starts with a Data-driven Distributionally Robust Optimization method utilizing a Wasserstein metric-based ambiguity set to determine the active and reactive power adjustments required for effective voltage control.

The study in¹⁷ introduces an innovative framework for smart homes aimed at enhancing energy consumption and production efficiency, ultimately reducing costs and bolstering grid reliability. This framework integrates the scheduling of controllable appliances and renewable energy sources while addressing uncertainties related to energy production, dynamic market prices, and variable household loads. Meanwhile, research in¹⁸ explores models for designing a demand-side management program leveraging advancements in cloud computing technology. The proposed model relies on the integration of storage units, enabling it to not only address variability in renewable energy production but also optimize energy reserves and computing capacity to achieve minimal operational costs. Work in¹⁹ introduces a model for determining the optimal size and location of renewable virtual power plants, integrated with a hydrogen storage system. It forms a key component of a multi-objective energy management framework tailored for an active distribution network.

Table 1 provides a streamlined summary of the key initiatives and discoveries discussed in the research background.

Table 1 Research summary.

Research gaps and contributions

Existing research highlights numerous shortcomings in energy management within distribution networks. A significant gap lies in the predominant focus of most studies^{3,4,5,6,7,8,9,10,11} on economic and operational indicators. While these are important, distribution networks often include both technical and economic parameters. Improving one indicator may not necessarily lead to enhancements in others. Under such conditions, issues like overvoltage can arise, potentially increasing power losses. Addressing this challenge requires incorporating multiple indices into energy management frameworks. Typically, distribution networks feature radial structures, where voltage magnitude tends to drop at the end of feeders. During instances of increased load, this can lead to voltage collapse. Therefore, managing energy in distribution networks necessitates considering voltage security limits, a factor only explored in a limited number of studies. Another notable deficiency is the insufficient acknowledgment of co-locating renewable resources and storage systems, as most research fragments these elements across various buses within the network. However, when such components are co-located to form a VPP, the network’s economic and technical performance can be significantly enhanced^{3,4,5,6,7,8,9,10,11}. For example, a renewable VPP integrated with a storage system could regulate its output power effectively through hydrogen storage alongside PV and BU. While batteries are commonly utilized as energy storage solutions in integrated energy systems throughout much of the research^{3,4,5,6,7,8,9,10,11}, they possess limitations such as short lifespans, high installation costs, and challenges in achieving larger capacity^5,6,7,8. This makes hydrogen storage a viable alternative due to its long lifespan, lower installation costs, and commendable efficiency. Yet, incorporating hydrogen storage in conjunction with PVs and BUs has received scant attention. Another critical research gap pertains to energy management in VPPs within distribution networks. Many studies fail to address the importance of optimal location and sizing of VPPs, these are factors crucial for maximizing effectiveness. The integration of VPP planning and operation within distribution networks remains underexplored, despite its potential to yield considerable benefits. Furthermore, renewable power production is inherently uncertain, causing discrepancies between day-ahead and real-time operations. In real-time scenarios, this inconsistency could disrupt the balance between production and consumption, thereby exposing a lack of system flexibility. Hydrogen storage offers a promising solution by enhancing VPP flexibility; however, the application of flexibility models in VPPs has been infrequently addressed in existing research. The critical research gaps in the field of VPPs can be summarized as follows: (1) a lack of emphasis on effective VPP planning, specifically identifying optimal grid locations and determining appropriate resource and storage device sizes; (2) insufficient exploration of hydrogen storage devices’ potential to enhance VPP performance and grid operations; (3) limited efforts in modeling diverse economic, technical, and environmental parameters; (4) underdeveloped strategies to mitigate power fluctuations from renewable sources, thus improving system flexibility.

Based on the limitations stated in the research background, the research questions are as follows: (1) Is the location of virtual power plants capable of improving the technical, economic, and environmental indicators in the distribution network? (2) Is the optimal size of energy resources and storage devices in VPP capable of improving the efficiency of this system in addition to improving the economic conditions of VPP? (3) Can the use of renewable solar and bio-waste resources along with hydrogen storage devices play a worthy role in improving the technical, economic, and environmental conditions of the distribution network?

This research aims to fill significant gaps in existing studies by introducing a comprehensive framework for planning and energy management within renewable VPPs equipped with hydrogen storage systems operating in active distribution networks. The approach utilizes a bi-level optimization structure to address the outlined challenges, as depicted in Fig. 1. At the upper level, a multi-objective optimization problem is established to align with the DSO priorities. These priorities encompass technological concerns such as ensuring the operational and voltage security of the network, alongside broader environmental and economic objectives. The model integrates operational constraints, voltage security criteria, and AC power flow equations pertinent to the distribution framework. The lower level concentrates on managing the operational and planning aspects of integrated systems, including hydrogen storage units, bio-waste units (BU), and photovoltaic (PV) installations. The objective function emphasizes minimizing VPP planning costs while meeting constraints derived from models of the BU, PV systems, and hydrogen storage. Additionally, it incorporates a flexibility framework linked to the VPP’s operation. To convert this bi-level optimization into a single-level formulation, the study utilizes the Karush–Kuhn–Tucker (KKT) conditions. A fuzzy decision-making framework is then applied to harmonize technical, economic, and environmental factors, delivering a balanced solution optimized for network performance. The unified model results in a nonlinear optimization problem. To solve this intricate nonlinear problem, the research employs the Red Panda Optimization (RPO) algorithm, a state-of-the-art computational method designed for superior efficiency. Furthermore, the framework incorporates Stochastic Based Scenario Optimization (SBSO) to account for uncertainties such as load variability, fluctuating energy prices, and inconsistency in renewable energy supply. Key contributions, developments and groundbreaking aspects of this proposed methodology include:

• Energy management within the distribution network can be significantly enhanced through the planning of an integrated PV, BU, and hydrogen storage system. This approach is designed to optimize key network indices, including operational efficiency, voltage security, economic performance, and environmental sustainability.

• To achieve this, the simultaneous modeling of operational, economic, and environmental indicators of the distribution network as a multi-objective optimization problem becomes vital. By incorporating voltage security constraints, this method effectively addresses the diverse goals of the DSO, providing a comprehensive framework for informed decision-making.

• Integrating a hydrogen storage system with PVs and BUs is a crucial step for mitigating fluctuations in the power output of both components. This integration enhances the overall responsiveness and stability of the energy system, allowing for improved regulation of network indices.

• The objectives of network operation and VPP planning, encompassing placement and sizing, can be efficiently modeled using a bi-level optimization approach. The proposed optimization challenges are resolved through the RPO algorithm, ensuring precise outcomes.

• Furthermore, the flexibility of VPPs is significantly augmented by the adoption of hydrogen storage systems, offering greater adaptability and resilience in distributed energy systems while addressing dynamic operational generations.

Figure 1 illustrates the comprehensive steps involved in the modeling process for VPPs planning, uncertainties, and the subsequent solution framework. As depicted, the developed approach adopts a bi-level optimization structure. In the upper-level problem, the energy management of the smart distribution network is formulated, incorporating VPPs into the model. Meanwhile, the lower-level problem captures the planning and operation of renewable resources, solar and bio-waste, as well as hydrogen storage systems, all within the VPP framework. To address these interconnected problems, the methodology necessitates transforming the bi-level structure into a single-level model. Given the convex nature of the lower-level formulation and its dependency on key variables required by the upper-level problem, the paper employs the KKT conditions to facilitate this single-level model extraction. Following the model simplification, uncertainties are tackled through a scenario-based stochastic optimization process. Finally, the optimal solution is derived using the RPO algorithm, ensuring robust decision-making in the presence of variability and complex constraints.

The second section focuses on modeling the proposed strategy through a multi-objective optimization framework, followed by an explanation of handling uncertainties using stochastic optimization techniques. Section 3 introduces the RPO methodology employed to address the outlined problem. Comprehensive analyses of numerical outcomes derived from various case studies are detailed in Sect. 4. Lastly, the study’s findings are summarized and presented in Sect. 5.

Fig. 1

Flowchart of bi-level flexible VPP planning in the smart distribution network.

Scheme model

Bi-level VPPs planning

This section delves into the energy scheduling strategies of the distribution network, particularly within the framework of planning a renewable VPP integrated with a hydrogen storage system. At the upper level, the model focuses on evaluating the operational, security, financial, and environmental objectives of the distribution system operator. Meanwhile, the lower level formulation incorporates a siting and sizing strategy to optimize system placement and capacity. The detailed structure of this plan is outlined as follows:

$$\:\text{min}F={v}_{1}Cost+{v}_{2}EL+{v}_{3}EM$$

(1)

Subject to:

$$\:Cost=\sum\:_{t,w}{}_{w}{p}_{t,w}{P}_{b=r,t,w}^{DS}$$

(2)

$$\:EL=\sum\:_{b,t,w}{}_{w}{(P}_{b,t,w}^{DS}+{P}_{b,t,w}^{I}-{P}_{b,t,w}^{C})$$

(3)

$$\:EM=\sum\:_{t,w}{}_{w}(CO2+SO2+NOX){P}_{b=r,t,w}^{DS}$$

(4)

$$\:{P}_{b,t,w}^{DS}+{P}_{b,t,w}^{I}+\sum\:_{j}{A}_{b,j}{P}_{b,j,t,w}^{L}={P}_{b,t,w}^{C}\:\:\:\:\:\:\:\forall\:b,t,w$$

(5)

$$\:{Q}_{b,t,w}^{DS}+\sum\:_{j}{A}_{b,j}{Q}_{b,j,t,w}^{L}={Q}_{b,t,w}^{C}\:\:\:\:\:\:\:\forall\:b,t,w$$

(6)

$$\:{P}_{b,j,t,w}^{L}={G}_{b,j}^{L}{\left({V}_{b,t,w}\right)}^{2}-{V}_{b,t,w}{V}_{j,t,w}\left\{{G}_{b,j}^{L}\text{cos}\left({\phi\:}_{b,t,w}-{\phi\:}_{j,t,w}\right)+{B}_{b,j}^{L}\text{sin}\left({\phi\:}_{b,t,w}-{\phi\:}_{j,t,w}\right)\right\}\:\:\:\:\forall\:b,j,t,w$$

(7)

$$\:{Q}_{b,j,t,w}^{L}={-B}_{b,j}^{L}{\left({V}_{b,t,w}\right)}^{2}+{V}_{b,t,w}{V}_{j,t,w}\left\{{B}_{b,j}^{L}\text{cos}\left({\phi\:}_{b,t,w}-{\phi\:}_{j,t,w}\right)-{G}_{b,j}^{L}\text{sin}\left({\phi\:}_{b,t,w}-{\phi\:}_{j,t,w}\right)\right\}\:\:\:\:\forall\:b,j,t,w$$

(8)

$$\:{\phi\:}_{b,t,w}=0\:\:\:\:\:\:\forall\:b=r,\:t,w$$

(9)

$$\:{V}_{b,t,w}=1\:\:\:\:\:\:\forall\:b=r,\:t,w$$

(10)

$$\:{{V}_{b}^{min}\le\:V}_{b,t,w}{V}_{b}^{max}\:\:\:\:\:\:\forall\:b,\:t,w$$

(11)

$$\:\sqrt{{{(P}_{b,j,t,w}^{L})}^{2}+{{(Q}_{b,j,t,w}^{L})}^{2}}\le\:{S}_{b,j}^{L,max}\:\:\:\:\:\:\forall\:b,j,t,w$$

(12)

$$\:\sqrt{{{(P}_{b,t,w}^{DS})}^{2}+{{(Q}_{b,t,w}^{DS})}^{2}}\le\:{S}_{b}^{DS,max}\:\:\:\:\:\:\forall\:b=r,t,w$$

(13)

$$\:{\left({V}_{p-1,t,w}\right)}^{4}-4{\left({V}_{p-1,t,w}\right)}^{4}\left\{{R}_{p,p-1}^{L}{P}_{p,p-1,t,w}^{L}+{X}_{p,p-1}^{L}{Q}_{p,p-1,t,w}^{L}\right\}-4{\left({X}_{p,p-1}^{L}{P}_{p,p-1,t,w}^{L}-{R}_{p,p-1}^{L}{Q}_{p,p-1,t,w}^{L}\right)}^{2}\ge\:{WSI}^{min}$$

(14)

$$\:{P}_{b,t,w}^{I}\in\:\{\text{min}PlC=\sum\:_{b}{(IC}_{b}^{PV}{S}_{b}^{PV}+{IC}_{b}^{BU}{S}_{b}^{BU}+{IC}_{b}^{HS}{S}_{b}^{HS})$$

(15)

$$\:{P}_{b,t,w}^{I}={P}_{b,t,w}^{PV}+{P}_{b,t,w}^{BU}+\left({P}_{b,t,w}^{DCH}-{P}_{b,t,w}^{CH}\right):\:{}_{b,t,w}^{I}\:\:\:\:\:\forall\:b,t,w$$

(16)

$$\:{P}_{b,t,w}^{PV}={\mu\:}_{t,w}^{PV}{S}_{b}^{PV}\::\:{}_{b,t,w}^{PV}\:\:\:\:\forall\:b,t,w$$

(17)

$$\:0\le\:{S}_{b}^{PV}\le\:{P}_{b}^{PV,max}\::\:{\underset{\_}{\theta\:}}_{b}^{PV},\:{\overline{\theta\:}}_{b}^{PV}\:\:\:\:\forall\:b,t,w$$

(18)

$$\:{P}_{b,t,w}^{BU}={\mu\:}_{t,w}^{BU}{S}_{b}^{BU}\::\:{}_{b,t,w}^{BU}\:\:\:\:\:\forall\:b,t,w$$

(19)

$$\:0\le\:{S}_{b}^{BU}\le\:{P}_{b}^{BU,max}\:\::\:{\underset{\_}{\theta\:}}_{b}^{BU},\:{\overline{\theta\:}}_{b}^{BU}\:\:\:\:\forall\:b,t,w$$

(20)

$$\:{0\le\:P}_{b,t,w}^{DCH}\le\:{S}_{b}^{HS}/{\tau\:}_{dch}\:\:\::\:{\underset{\_}{\theta\:}}_{b,t,w}^{DCH},\:{\overline{\theta\:}}_{b,t,w}^{DCH}\:\:\:\:\forall\:b,t,w$$

(21)

$$\:{0\le\:P}_{b,t,w}^{CH}\le\:{S}_{b}^{HS}/{\tau\:}_{ch}\:\::\:{\underset{\_}{\theta\:}}_{b,t,w}^{CH},\:{\overline{\theta\:}}_{b,t,w}^{CH}\:\:\:\:\:\forall\:b,t,w$$

(22)

$$\:{P}_{b,t,w}^{CH}{P}_{b,t,w}^{DCH}=0\:\:\::\:{}_{b,t,w}^{HS}\:\:\:\:\:\:\:\:\:\:\forall\:b,t,w$$

(23)

$$\:{E}_{b,t,w}=\alpha\:\times\:{S}_{b}^{HS}+\sum\:_{h=1}^{t}({}^{CH}{P}_{b,h,w}^{CH}-\frac{1}{{}^{DCH}}{P}_{b,h,w}^{DCH})\::\:{}_{b,t,w}^{E}\:\:\:\:\:\forall\:b,t,w$$

(24)

$$\:{\beta\:\times\:S}_{b}^{HS}\le\:{E}_{b,t,w}\le\:{S}_{b}^{HS}\:\:\::\:{\underset{\_}{\theta\:}}_{b,t,w}^{E},\:{\overline{\theta\:}}_{b,t,w}^{E}\:\:\:\:\:\:\:\forall\:b,t,w$$

(25)

$$\:0\le\:{S}_{b}^{HS}\le\:{E}_{b}^{max}\::\:{\underset{\_}{\theta\:}}_{b}^{HS},\:{\overline{\theta\:}}_{b}^{HS}\:\:\:\:\:\:\:\forall\:b$$

(26)

$$\:{-\varDelta\:F\le\:P}_{b,t,w}^{I}-{P}_{b,t,w=1}^{I}\le\:\varDelta\:F\::\:{\underset{\_}{\theta\:}}_{b,t,w}^{F},\:{\overline{\theta\:}}_{b,t,w}^{F}\:\:\:\:\:\:\:\forall\:b,t,w,\}$$

(27)

Upper level model

This section follows the framework outlined in Eq. (1) through (14). Specifically, Eq. (1) defines the objective function of the proposed strategy, aiming to reduce the weighted aggregate of the operation costs, energy losses, and emission levels within the distribution network. Evaluating the economic performance of a distribution network necessitates the use of an operating cost function that optimizes energy efficiency. This function is designed to ensure consumer supply at minimal energy costs. The operating cost of the network (Cost), formally outlined in Eq. (2)¹, serves as a key parameter for modeling the financial aspects of energy procurement from the upstream network. Specifically, this cost encapsulates the network’s total energy consumption expenses related to passive loads and the cost tied to energy losses. As such, minimizing this cost becomes a strategic goal for the DSO, making it an integral part of its operational objectives. In the distribution network model, the total energy consumption and associated losses are equivalent to the energy sourced from the distribution substation connected to the upstream grid. Consequently, Eq. (2) incorporates the power utilized by this substation. The energy cost itself is calculated as the product of energy price and consumption, forming the basis on which Eq. (2) is structured. Each consumer within the network utilizes energy at a specific price, necessitating inclusion of the energy price within Eq. (2). However, it is important to emphasize that Eq. (2) does not represent a market model for the distribution network. Furthermore, the operational costs of VPPs are integrated into this cost function, as described in Eq. (2). When VPPs operate as consumers or generators, the energy flow through the distribution substation adjusts accordingly. In such scenarios, the total energy passing through the substation equals the sum of the energy consumed by passive loads and losses, with an addition or deduction based on the energy contributions of VPPs in their respective roles. As a result, Eq. (2) also accounts for the costs or revenues associated with VPP operations, whether they are consuming energy or generating and selling it, thereby providing a comprehensive evaluation of network costs under varying operational conditions. Equation (3) presents the model for anticipated energy losses (EL), defined as the difference between the energy consumed and the energy generated. Furthermore, Eq. (4) quantifies the network’s emissions (EM) arising from the energy supplied by the upstream network. This study incorporates environmental emissions caused by SO₂, NO_X, and CO₂ as key factors¹⁹. The weighted sum of Cost, EL, and EM constitutes the objective function of the problem (F), with the weight coefficients denoted as v1, v2, and v3. These coefficients collectively sum to 1, ensuring that adjustments to their values influence the outcomes of the Cost, EL, and EM functions. By graphing these functions in three-dimensional space, corresponding points on the Pareto front of the proposed scheme can be identified²⁰. To determine the optimal compromise solution among conflicting objectives, the research employs a fuzzy decision-making framework. This approach calculates the minimum (F^min) and maximum (F^max) values for each function based on three scenarios: v1 = 1, v2 = 1, and v3 = 1. Subsequently, a linear membership function (fl.) is defined, where fl. equals zero if the function value exceeds F^max or one if it is lower than F^min. Otherwise, fl. is given by (f – F^max)/(F^min – F^max), where f represents the function value. The same process is applied to the Cost, EL, and EM functions to identify the smallest value of fl. Adjustments are then made to the weight coefficients, and both σ and fl. values for respective functions are recalculated. This iterative procedure is repeated for every point on the Pareto front. Ultimately, the point with the highest σ value is identified as the optimal compromise solution²⁰. The constraints (5)–(13) define the parameters governing the distribution network model. Equation (5) to (10) illustrate the AC power flow within the network. Specifically, Eqs. (5) and (6) represent active and reactive power balance at various buses. Equations (7) and (8) calculate the active and reactive power flow across distribution lines. Additionally, Eqs. (9) and (10) specify angle and voltage magnitude at the reference bus. Constraints (11) through (13) account for operational limitations of the network. Equation (11) establishes voltage magnitude restrictions at various buses, while Eqs. (12) and (13) impose apparent power limits across distribution lines and substations^{19– 20}. The analysis assumes a single distribution substation connected to an upstream network and situated at the reference bus. Consequently, in Eqs. (2), (4), and (13), the bus index (b) equates to the reference bus index (r). In constraint (14), the voltage security limit is expressed. In this paper, the worst security index (WSI) is used to evaluate voltage security²⁰. WSI is calculated on the left side of Eq. (14). It has a value between zero and one. Its value of zero means voltage collapse in the network. But if WSI has a value of 1, the network operates in no-load condition. Therefore, it is necessary that as in constraint (14), the WSI amount is greater than a certain value such as minimum WSI value (WSI^min)²⁰.

The primary model for analyzing power system behavior is the AC power flow, which serves as the foundation of this study. In particular, the focus is on the distribution network. By utilizing the AC power flow model, this approach ensures more precise and reliable calculations of network voltage and power, delivering results with greater accuracy and credibility.

Lower-level model

This section is structured around equations (15) through (27), outlining the mathematical framework for VPP planning and operation. The core objective is defined in Eq. (15), which seeks to minimize the total VPP planning costs. These costs consist of the expenses associated with PV, BU and hydrogen storage (HS). The cost of each element is calculated as the product of its unit construction cost and capacity. Equation (16) through (26) describe the operational and planning constraints for the integrated PV-BU-hydrogen storage system. Equation (16) determines the flow of electricity injected into or withdrawn from the distribution network, based on the power outputs of PV, BU, and HS components. Equation (17) calculates the electricity generated by the PV system, which is determined by multiplying its capacity by its generation rate². PV size limitations are addressed in Eq. (18). The planning and operation model for BU is detailed in Eqs. (19) and (20). Equation (19) computes BU’s active power output, which corresponds to its power rate multiplied by its optimal capacity. BU capacity constraints are defined in Eq. (20). For hydrogen storage, equations (21) through (26) delineate its operational²¹. Constraints (21) and (22) specify the maximum charge and discharge limits of HS. To prevent simultaneous charging and discharging, Eq. (23) ensures that when HS operates in charging mode, its discharge power remains zero, and vice versa. Equation (24) calculates the stored energy in HS as the sum of its initial energy plus the energy accrued during charging cycles, minus the energy discharged. Constraint (25) imposes limitations on the maximum storable energy, which determines the appropriate HS size or capacity. The capacity or size of HS is formulated in (26). In fact, Eq. (25) refers to the issue of what size or capacity of hydrogen storage device should be installed in VPP. Finally, constraint (27) introduces the VPP flexibility model. Since renewable energy sources like PV entail inherent uncertainties in power generation, VPP’s output varies across scenarios. To enhance operational flexibility, VPP power should be stabilized across these scenarios. Constraint (27) achieves this by incorporating a flexibility tolerance parameter denoted as ΔF; a value approaching zero signifies near-perfect flexibility. Dual variables κ and θ are used to represent lower-level constraints within this optimization framework.

The HS device incorporates three key components: a hydrogen tank, an electrolyzer, and a fuel cell. During the charging phase, the electrolyzer converts received electrical energy into hydrogen, which is subsequently stored in the tank. Conversely, in the discharging phase, the fuel cell processes hydrogen extracted from the tank to generate electrical energy, which is then delivered to the VPP. As a result, the active power during the charging phase corresponds to the electrolyzer’s output, while the active power in the discharging phase aligns with the fuel cell’s output. The energy quantified in Eq. (24) reflects the amount of hydrogen stored in the tank. In this context, the terms η^CH and η^DCH signify the efficiency of the electrolyzer and fuel cell, respectively. The hydrogen charging and discharging rates are expressed as $\:{S}_{b}^{HS}/{\tau\:}_{ch}$ and $\:{S}_{b}^{HS}/{\tau\:}_{dch}$, respectively.

In the proposed approach, it is presumed that the optimal sizing of resources and storage facilities, along with the ideal placement of the VPPs, is determined and implemented within the initial year. Furthermore, the associated costs are assumed to be incurred at the outset of the planning phase. As a result, based on Eq. (15), only the bus index (b) is necessary, while the time index (t) becomes irrelevant. Consequently, the planning problem within this framework is not time-sensitive, eliminating the need for establishing a specific planning horizon. However, the exploitation problem is inherently time-dependent, thus requiring the definition of an exploitation horizon tailored to this aspect.

In the upper-level problem, the active power of the virtual power plant (P^I) serves as the key decision variable. Meanwhile, in the lower-level formulation, the decision variables include the capacity of the photovoltaic system, bio-waste unit, and hydrogen storage system (denoted as S^PV, S^BU, and S^HS, respectively), along with the active power of the hydrogen storage system in both charging (P^CH) and discharging (P^DCH) modes.

SBSO-based uncertainty model

This study identifies several key parameters such as load, P^C and Q^C, energy price (ρ), and power production rates for PV and BU systems (µ^PV and µ^BU) as uncertain parameters within Eqs. (1)–(27). To effectively address the prediction errors stemming from these uncertainties, the research employs the SBSO framework. The adopted SBSO methodology integrates a hybrid approach combining the Kantorovich technique and RWM²². Initially, this framework generates multiple scenarios utilizing the RWM. For load, power generation rate of BU, and energy price parameters in each scenario, a normal probability distribution function (PDF) is applied to evaluate their likelihood. Simultaneously, the beta probability distribution function determines the probability values for PV power production rates. Consequently, the probability of each generated scenario (π⁰) is calculated by multiplying the probabilities associated with all uncertain variables. The Kantorovich approach is then introduced as a scenario reduction mechanism. This step refines the set of generated scenarios by selecting those that exhibit higher similarity to one another. Next, Eqs. (1)–(27) are solved using only the reduced scenario set. Following this, the adjusted probability of each selected scenario is computed by dividing its original π⁰ value by the sum of π⁰ values across all chosen scenarios. Comprehensive details regarding this methodology are provided in²². For ease of implementation, Algorithm 1 outlines the complete SBSO procedure.

Algorithm. 1

SBSO process.

Solution approach

Single level formulation

The optimization problem at hand represents a convex nonlinear challenge situated within a bi-level framework. To tackle its hierarchical structure, the Karush-Kuhn-Tucker (KKT) method is employed, enabling the transformation of the original bi-level problem into a single-level formulation²³. This process commences with the construction of a Lagrangian function (L) that captures the essence of the lower-level problem. The Lagrangian incorporates the objective function along with penalty terms that account for the associated constraints. Penalty functions are defined as follows: for an equality constraint (a = b), the penalty is expressed as κ.(a − b), whereas for an inequality constraint (a ≤ b), it takes the form θ.max(0, a − b)²⁴. Here, κ and θ serve as dual variables corresponding to the lower-level constraints. Notably, κ is unrestricted across (-∞, +∞), while θ is bound to [0, +∞]. Once this Lagrangian is formulated, the transformed single-level model integrates the objectives of the upper-level problem alongside a set of derived KKT constraints from the lower-level counterpart. These constraints arise due to the zero-gradient condition of the Lagrangian function with respect to both its primal and dual variables. Specifically, this condition brings forth two critical elements: the restoration of the original lower-level constraints, such as those described by Eqs. (15)-(27), and an additional requirement characterized by θ.(a − b) = 0. Moreover, further KKT constraints emerge from setting the derivative of the Lagrangian function with respect to each primal variable equal to zero²³. The primary variables within the lower-level problem encompass diverse parameters including P^I, P^PV, P^BU, P^CH, P^DCH, S^PV, S^BU, S^HS, and E. These play a central role in shaping both the constraints and outcomes of the unified single-level optimization model. In general, the single-level model for a bi-level problem includes the upper level formulation and the KKT model for the lower level problem. The KKT model for a problem includes the constraints resulting from the equality of the derivative of the Lagrange function to the primary and dual variables with zero. Below is the detailed structure of the resulting single-level model:

$$Upper{\text{ }}level{\text{ }}\bmod e,{\text{ }}i.e.{\text{ }}Eqs.{\text{ }}\left( 1 \right) - \left( {14} \right)$$

(28)

Lover level constraints, i.e. Equations (16)-(27):

$$\partial L/\partial \theta {\text{ }} = {\text{ }}0$$

(29)

$$\:\left({S}_{b}^{PV}-{P}_{b}^{PV,max}\right){\overline{\theta\:}}_{b}^{PV}=0:\frac{L}{{\overline{\theta\:}}_{b}^{PV}}=0\:\:\:\forall\:b$$

(30)

$$\:{S}_{b}^{PV}{\underset{\_}{\theta\:}}_{b}^{PV}=0\::\frac{L}{{\underset{\_}{\theta\:}}_{b}^{PV}}=0\:\:\:\forall\:b$$

(31)

$$\:\left({S}_{b}^{BU}-{P}_{b}^{BU,max}\right)\:\:{\overline{\theta\:}}_{b}^{BU}=0\:\::\frac{L}{{\overline{\theta\:}}_{b}^{BU}}=0\:\:\forall\:b$$

(32)

$$\:{S}_{b}^{BU}{\underset{\_}{\theta\:}}_{b}^{BU}=0\::\frac{L}{{\underset{\_}{\theta\:}}_{b}^{BU}}=0\:\:\:\:\forall\:b$$

(33)

$$\:\left({P}_{b,t,w}^{DCH}-{S}_{b}^{HS}/{\tau\:}_{dch}\right)\:\:{\overline{\theta\:}}_{b,t,w}^{DCH}=0\::\frac{L}{{\overline{\theta\:}}_{b,t,w}^{DCH}}=0\:\:\:\forall\:b,t,w$$

(34)

$$\:{P}_{b,t,w}^{DCH}{\underset{\_}{\theta\:}}_{b,t,w}^{DCH}=0\::\frac{L}{{\underset{\_}{\theta\:}}_{b,t,w}^{DCH}}=0\:\:\:\:\forall\:b,t,w$$

(35)

$$\:\left({P}_{b,t,w}^{CH}-{S}_{b}^{HS}/{\tau\:}_{ch}\right)\:{\overline{\theta\:}}_{b,t,w}^{CH}=0:\frac{L}{{\overline{\theta\:}}_{b,t,w}^{CH}}=0\:\:\:\:\forall\:b,t,w$$

(36)

$$\:{P}_{b,t,w}^{CH}{\underset{\_}{\theta\:}}_{b,t,w}^{CH}=0\::\frac{L}{{\underset{\_}{\theta\:}}_{b,t,w}^{CH}}=0\:\:\:\forall\:b,t,w$$

(37)

$$\:\left({E}_{b,t,w}-{S}_{b}^{HS}\right)\:\:{\overline{\theta\:}}_{b,t,w}^{E}=0\:\::\frac{L}{{\overline{\theta\:}}_{b,t,w}^{E}}=0\:\:\:\:\:\forall\:b,t,w$$

(38)

$$\:\left({\beta\:\times\:S}_{b}^{HS}-{E}_{b,t,w}\right)\:{\underset{\_}{\theta\:}}_{b,t,w}^{E}=0\:\::\frac{L}{{\underset{\_}{\theta\:}}_{b,t,w}^{E}}=0\:\:\:\:\:\:\forall\:b,t,w$$

(39)

$$\:\left({S}_{b}^{HS}-{E}_{b}^{max}\right)\:\:{\overline{\theta\:}}_{b}^{HS}=0\::\frac{L}{{\overline{\theta\:}}_{b}^{HS}}=0\:\:\:\:\:\:\:\forall\:b$$

(40)

$$\:{S}_{b}^{HS}\:{\underset{\_}{\theta\:}}_{b}^{HS}=0\::\frac{L}{{\underset{\_}{\theta\:}}_{b}^{HS}}=0\:\:\:\:\:\:\:\forall\:b$$

(41)

$$\:{(-\varDelta\:F-(P}_{b,t,w}^{I}-{P}_{b,t,w=1}^{I}){\underset{\_}{)\theta\:}}_{b,t,w}^{F}=0\::\frac{L}{{\underset{\_}{\theta\:}}_{b,t,w}^{F}}=0\:\:\:\:\:\:\:\:\forall\:b,t,w,\}$$

(42)

$$\:{(P}_{b,t,w}^{I}-{P}_{b,t,w=1}^{I}-\varDelta\:F\:)\:{\overline{\theta\:}}_{b,t,w}^{F}=0\::\frac{L}{{\overline{\theta\:}}_{b,t,w}^{F}}=0\:\:\:\:\:\:\forall\:b,t,w,\}$$

(43)

$$\:\:{}_{b,t,w}^{I}-{\underset{\_}{\theta\:}}_{b,t,w}^{F}+\:{\overline{\theta\:}}_{b,t,w}^{F}+\sum\:_{w}\left({\underset{\_}{\theta\:}}_{b,t,w}^{F}-\:{\overline{\theta\:}}_{b,t,w}^{F}\right)=0\::\frac{L}{{P}_{b,t,w}^{I}}=0\:\:\:\:\:\:\:\forall\:b,t,w$$

(44)

$$\:\:{}_{b,t,w}^{PV}-{}_{b,t,w}^{I}=0\::\frac{L}{{P}_{b,t,w}^{PV}}=0\:\:\:\:\:\:\:\forall\:b,t,w$$

(45)

$$\:\:{}_{b,t,w}^{BU}-{}_{b,t,w}^{I}=0\::\frac{L}{{P}_{b,t,w}^{BU}}=0\:\:\:\:\:\:\:\forall\:b,t,w,$$

(46)

$$\:\:{\overline{\theta\:}}_{b,t,w}^{DCH}-{\underset{\_}{\theta\:}}_{b,t,w}^{DCH}+{P}_{b,t,w}^{CH}{}_{b,t,w}^{HS}-{}_{b,t,w}^{I}+\frac{1}{{}^{DCH}}\sum\:_{h=t}^{24}{}_{b,h,w}^{E}=0\::\frac{L}{{P}_{b,t,w}^{DCH}}=0\:\:\:\:\:\:\:\forall\:b,t,w$$

(47)

$$\:\:{\overline{\theta\:}}_{b,t,w}^{CH}-{\underset{\_}{\theta\:}}_{b,t,w}^{CH}+{P}_{b,t,w}^{DCH}{}_{b,t,w}^{HS}+{}_{b,t,w}^{I}-{}^{CH}\sum\:_{h=t}^{24}{}_{b,h,w}^{E}=0\::\frac{L}{{P}_{b,t,w}^{CH}}=0\:\:\:\:\:\:\:\forall\:b,t,w$$

(48)

$$\:\:{\overline{\theta\:}}_{b,t,w}^{E}-{\underset{\_}{\theta\:}}_{b,t,w}^{E}+{}_{b,t,w}^{E}=0\::\frac{L}{{E}_{b,t,w}}=0\:\:\:\:\:\:\:\forall\:b,t,w$$

(49)

$$\:\:{\overline{\theta\:}}_{b}^{PV}-{\underset{\_}{\theta\:}}_{b}^{PV}-\sum\:_{t,w}{}_{t,w}^{PV}{}_{b,t,w}^{PV}={IC}_{b}^{PV}\::\frac{L}{{S}_{b}^{PV}}=0\:\:\:\:\:\:\:\forall\:$$

(50)

$$\:\:{\overline{\theta\:}}_{b}^{BU}-{\underset{\_}{\theta\:}}_{b}^{BU}-\sum\:_{t,w}{}_{t,w}^{BU}{}_{b,t,w}^{BU}={IC}_{b}^{BU}\::\frac{L}{{S}_{b}^{BU}}=0\:\:\:\:\:\:\:\forall\:$$

(51)

$$\:\:{\overline{\theta\:}}_{b}^{HS}-{\underset{\_}{\theta\:}}_{b}^{HS}+\:\sum\:_{t,w}\left(-{\overline{\theta\:}}_{b,t,w}^{E}+{\underset{\_}{\theta\:}}_{b,t,w}^{E}-{}_{b,t,w}^{BU}-\frac{{\overline{\theta\:}}_{b,t,w}^{DCH}}{{\tau\:}_{dch}}-\frac{{\overline{\theta\:}}_{b,t,w}^{CH}}{{\tau\:}_{ch}}\right)={IC}_{b}^{HS}\::\frac{L}{{S}_{b}^{HS}}=0\:\:\:\:\:\:\:\forall\:$$

(52)

$$\:\:\theta\:\ge\:0,\:\in\:(-\infty\:,+\infty\:)$$

(53)

Problem solving by RPO

The problem described by Eq. (1) to (27) is modeled as a nonlinear optimization challenge. To address this, the paper employs the RPO algorithm, a robust method designed to tackle complex engineering optimization problems with high reliability in obtaining optimal solutions²⁵. Initially, RPO generates N (population size) random values for the decision variables, which include S^PV, S^BU, S^HS, P^CH, P^DCH, κ, and θ. These values are assigned based on their governing equations, such as (18), (20), (26), (21), and (22), within the predefined limits of (-∞, +∞) and [0, +∞), respectively. Once the decision variables are initialized for N population members, the dependent variables are computed accordingly. In this model (defined by Eqs. (1)-(27)), all variables apart from the decision variables are classified as dependent. Specifically, P^I, P^PV, P^BU, and E are derived based on Eqs. (16), (17), (19), and (24). The remaining dependent variables are calculated using Eqs. (2)-(10), which describe the power flow model within the distribution network²⁶. To solve these constraints, the backward-forward power flow technique²⁶ is applied. Subsequently, the fitness function is evaluated for each of the N sets of decision variables. This function combines two components: the objective function derived from Eq. (1) and the penalty function addressing constraints specified in Eqs. (11)-(13), (23), (25), (27), θ.(a–b) = 0, and ∂L/∂s = 0. The term ∂L/∂s = 0 reflects the condition that the derivative of the Lagrangian function with respect to primal variables equals zero, as discussed in Sect. 3.1. Based on the optimal fitness values obtained in the current iteration, RPO updates the decision variables and recalculates the dependent variables along with the new fitness function. This iterative process continues until a convergence criterion is met. In this study, convergence is assumed to occur at a maximum of iterations (I_max).

Numerical results

Data

In general, the proposed scheme with the formulation presented in Sects. 2 and 3 has no limitations for implementation on different data of network, renewable sources and hydrogen storage devices. This section focuses on the IEEE 69-bus distribution network²⁷, which serves as the test network. The system operates with a base voltage of 12.66 kV and a base power of 1 MVA. Bus 1, designated as the reference bus, connects the distribution substation to the upstream network. The permissible voltage range spans from 0.9 p.u. to 1.05 p.u. Details about each bus’s peak active and reactive load, distribution line specifications, and substation parameters can be found in²⁷. To determine load levels at various hours, the peak load is scaled using a daily load factor curve. The anticipated load factor curve has been outlined in¹. Energy price at [1:00 7:00] and [17:00 22:00] is 16 $/MWh and 30 $/MWh, respectively. It is 24 $/MWh for other hours¹. The sum of emissions coefficients (CO2 + SO2 + NOX) is 767.4 ton/MW¹³. WSI^min is 0.8 p.u²⁰. This paper builds upon the model outlined in Eq. (1) through (27), presuming that a renewable virtual power plant (VPP) equipped with hydrogen storage (HS) can be installed at each bus. The proposed system design includes a maximum capacity of 0.4 MW for photovoltaic panels, 0.8 MW for bio-wate units, and 3 MWh for hydrogen storage units. For HS-related parameters, the α and β values are set at 0.1. The charging and discharging processes are designed to operate within a 3-hour timeframe, with respective efficiencies of 75% and 51%. The expected daily power generation profiles for PV panels and BU follow the curves provided in references^28,29. Regarding costs, installation expenses are quantified as follows: PV systems at 5.6 million dollars per MW, BU at 6.5 million dollars per MW, and HS units at 3 million dollars per MWh. These estimates stem from the same aforementioned references. Additionally, to ensure optimal flexibility conditions for VPP operations, the flexibility threshold (ΔF) is set to 0.02 MW.

Results and discussion

This section utilizes MATLAB to encode problems (1)–(27) corresponding to the data presented in Sects. 4 − 1. For uncertainty modeling, a standard deviation of 10% is assumed³⁰. A total of 2000 scenarios are generated using RWM, which are then reduced to 80 scenarios using Kantorovich’s method, a scenario reduction technique applied to the proposed problem. The solution algorithm is initialized with a population size of 80, and the maximum number of convergence iterations is set to 4000.

Analysis of DSO objectives

At a 100% load level, the Pareto front of the proposed strategy is visually depicted in Fig. 2. This figure illustrates the variations in the network operator’s objective functions, i.e. operation cost (Cost), energy losses (EL), and emissions (EM), under different weight coefficients. On the horizontal axis, weight coefficients are represented, while the vertical axis denotes the values of the objective functions. As observed, the changes in Cost, EL, and EM do not follow a uniform pattern; notably, minimizing costs results in an increase in EL values. To achieve cost reduction, combined PV-bio-waste-Hydrogen storage systems must inject substantial active power, which can lead to increased power and energy losses due to current flow from these systems toward the distribution substation. Based on Fig. 2 data, the minimal values for Cost, EL, and EM stand at $894.2, 1.12 MWh, and 21.3 tons respectively, while their maximums reach $3612.1, 2.61 MWh, and 38.4 tons. Consequently, the range of variation across these metrics is $2717.9 for Cost, 1.49 MWh for EL, and 17.1 tons for EM. Furthermore, decision-making for various load levels reveals an optimal compromise solution between Cost, EL, and EM, detailed in Fig. 3. At this compromise point (for a 100% load level), the three functions measure $1289.3 for Cost, 1.55 MWh for EL, and 24.9 tons for EM. Using a comparative analysis between Figs. 3 and 4, it becomes evident that the deviation of the compromise point values from their respective minima is significant: Cost increases by 14.5% ((1289.3–894.2)/2717.9), while EL and EM rise by 21% and 28.9%, respectively. This indicates that fuzzy decision-making effectively identifies a balanced solution where function values remain close to their minimal levels while achieving an optimal trade-off. Lastly, Fig. 3 illustrates a clear trend: values for Cost, EL, and EM escalate as load levels intensify. This is attributed to higher utilization of network load under specific operational scenarios, underscoring the impact of increased demand on these objectives.

Fig. 2

Scheme Pareto front for load level of 100%.

Fig. 3

Compromise solution for Cost, EL, and EM considering different load levels.

VPPs planning

This section delves into the planning and operational strategies for integrated PV-BU-HS systems. Figure 4 underscores the strategic placement of VPPs across the distribution network to meet objectives set by the Distribution System Operator. Six VPP installations are spaced along the feeder, with endpoints prioritizing network enhancements such as improved voltage profiles and energy loss reduction. Meanwhile, installations at the start or middle focus primarily on maximizing power injection, fostering economic and environmental gains. Key details regarding the sizes of PVs, BUs, and hydrogen storage systems within each VPP are also captured in Fig. 4. Smaller PV sizes are attributed to their intermittent operation, whereas BUs, characterized by continuous and higher energy output per unit capacity, are allocated more substantial proportions. Likewise, large PV and BU capacities correspond to larger HS sizes, whereas limited resource capacities result in scaled-down HS installations. VPPs at feeder endpoints typically feature smaller resources and HS capacities due to constraints posed by the distribution line’s limited capabilities. In contrast, locations at the feeder’s start or middle allow for larger setups due to the absence of such limitations. Cost analysis reveals significant disparities in planning expenditures across these installations, with VPPs at feeder ends incurring lower costs than those positioned near the beginning. The total setup cost for all VPPs amounts to $58.32 million, distributed accordingly: $13.08 M, $6.36 M, $9.72 M (twice), $12.43 M, and $7.01 M. Following this, Fig. 5 outlines the daily power generation profiles of PVs, BUs, and HSs at varying load levels. Figure 5a illustrates that PV power output aligns closely with findings in prior studies, showing potential deployment of full capacities under optimal conditions. For BUs, Fig. 5b confirms stable power generation throughout their operation period. Hydrogen storages exhibit a distinct charging mode between 1:00 and 16:00 and switch to discharging from 17:00 to 22:00, coinciding with peak electricity pricing times, which contributes substantially to reducing operational costs and environmental impact. Additionally, Fig. 5c highlights maximum power consumption by HSs between 1:00 and 7:00 during low electricity price intervals to further minimize expenses. From 8:00 to 16:00, HSs switch to discharge mode to mitigate overvoltage risks arising from peak PV production. Moving forward, Fig. 5d showcases the unified active power generation curve for integrated PV-BU-HS systems across a range of load levels. Derived using relation (16), this curve reflects the aggregate discharge power from PVs, BUs, and HSs after subtracting HS charging power. Notably, these systems exhibit power consumption only between 1:00 and 7:00 and shift to electricity generation during other hours. Early morning operation sees HSs act as primary energy consumers due to limited PV and BU electricity supply. Later in the day, these systems transition into a generation mode as demand surges. The energy profiles of PVs and BUs remain stable regardless of variations in network loads given their independence from demand fluctuations. However, HS activities intensify with increased load conditions, evident in both charge and discharge phases. Accordingly, the active power output from integrated systems escalates as load levels grow, enhancing both their consumption efficiency and electricity production capabilities throughout the day.

Fig. 4

Location of VPPs considering optimal size of sources and storages in smart distribution network for load level of 100%.

Fig. 5

Expected daily active power curve, (a) PVs, (b) BUs, (c) HSs, (d) VPPs for different load levels.

Network indices

This analysis assesses the technical, economic, and environmental performance of a distribution network by examining various metrics, including economic cost (Cost), environmental impact (EM), voltage security index (WSI), and operational indices such as energy loss (EL), maximum voltage drop (MVD), and maximum overvoltage (MOV). These indicators are presented in Fig. 6 for a full load level across different research scenarios. Three case studies are considered in the evaluation: Case I involves load distribution entirely dependent on the upstream network; Case II investigates a proposed photovoltaic (PV)-battery-only setup; and Case III explores an integrated PV-battery system combined with hydrogen storage. Figures 6a, b highlight that Case I leads to the highest values for Cost, EM, EL, and MVD, primarily because all energy is sourced from the upstream network, resulting in inefficient system performance. Additionally, Fig. 6c shows a noticeably low WSI for this scenario, suggesting weaker voltage security. Conversely, scenarios incorporating photovoltaic systems into the distribution network, Case II and Case III, display marked improvements across most indicators compared to Case I, with the exception of MOV, which shows a slight increase. Among these, Case III stands out for achieving the most substantial reductions in Cost, EL, EM, and MOV values. Operational performance is further investigated through the Peak Load Carrying Capacity (PLCC), outlined in Fig. 6a. This metric quantifies the maximum load the network can sustain based on daily load factor patterns as defined in¹. Both Case I and Case II maintain similar PLCC levels due to inactive PV systems during peak demand hours (20:00). However, Case III surpasses these cases by achieving a higher PLCC value, attributed to hydrogen storage injecting significant power into the system during peak hours (17:00–22:00). Furthermore, Case III enhances voltage security by delivering a higher WSI based on Fig. 6c. Nonetheless, bus 65 consistently experiences a limited voltage range across all scenarios despite these advancements.

Flexibility is examined in Table 2 by analyzing the power deviation of VPPs across different cases. In Case I, flexibility cannot be defined due to the absence of VPPs. Case II includes renewable resources exclusively within the VPP but demonstrates high power deviation, reflecting poor flexibility. Conversely, Case III exhibits minimal power deviation, indicating favorable flexibility status and better adaptability to operational changes.

Fig. 6

Value of the different indices of smart distribution network for load level of 100%.

Table 2 Flexibility situation of VPPs in the different cases.

Analysis of the convergence performance of the proposed scheme

This section assesses the convergence behavior of the proposed scheme in comparison with several optimization techniques, including the RPO, Krill Herd Optimization (KHO)³¹, Crow Search Algorithm (CSA)³², Artificial Bee Colony (ABC) algorithm³³, Differential Equation (DE) algorithm³⁴, Particle Swarm Optimization (PSO)³⁵, and Genetic Algorithm (GA)³⁶. Figure 7 presents the results of these comparisons, conducted using a population size of 80 and a maximum convergence iteration of 4000. The tuning parameters for each algorithm were determined based on references^{25,31,32,33,34,35,36}. Each algorithm executed the problem-solving process 30 times, after which the standard deviation (StD) of their final solutions was calculated. From the figure, the RPO algorithm achieved the lowest value for the DSO objective function and the VPP objective function. Its optimal solution corresponded to a convergence iteration (CI) of 1973, whereas all other algorithms required more than 2500 iterations. Furthermore, RPO exhibited superior computational efficiency, with a runtime of 5.6 min compared to over 7 min for the other algorithms. These results highlight RPO’s advantages as a solver, with better optimality and faster convergence. It is notable that the proposed scheme addresses a non-convex problem, where solvers typically find local optima rather than global ones³⁷. In such scenarios, selecting the solver that achieves the most optimal solution is crucial³⁸, as the proximity of its results to the global optimum is generally expected to be smaller³⁹. This behavior was observed for the proposed scheme in RPO. Additionally, RPO demonstrated the lowest standard deviation among the tested algorithms, indicating minimal dispersion in its final outputs. This further reinforces its reliability as an effective optimization method.

Fig. 7

Convergence situation in the different solvers.

Conclusion

This research delves into the energy management of a distribution network by examining the Distribution System Operator’s operational, security, financial, and environmental goals for planning and operating systems that integrate photovoltaic, bio-waste units, and hydrogen storage technologies. The study adopts a bi-level optimization framework. At the upper level, decisions are driven by the network’s optimal power distribution equations, combined with operational and security constraints, aiming to minimize a weighted sum of expenses, energy losses, and emission levels. At the lower level, the focus shifts to managing virtual power plants integrated with hydrogen storage systems. This level seeks to minimize planning costs while accounting for constraints tied to the operation and planning of PV systems, BU, and hydrogen storage units, along with exploiting VPP flexibility features. To address uncertainties, such as fluctuating load demands, PV generation variability, BU performance, and energy price changes, stochastic optimization techniques were employed. The integration of fuzzy decision-making helped pinpoint an optimal compromise solution, while the KKT conditions streamlined the formulation of the bi-level model into a single-level optimization approach. Numerical analyses showed that fuzzy decision-making led to a compromise scenario where operational costs, energy losses, and emissions were respectively 14%, 29%, and 21% higher than their absolute minimums. Further evaluation revealed that configurations relying solely on PV and BU enhanced outcomes like reduced maximum voltage deviations, emissions, operational costs, and energy losses but introduced severe over-voltage issues within the grid. Conversely, the inclusion of hydrogen storage in network operations yielded widespread improvements across operational efficiency, security metrics, environmental sustainability, and economic performance compared to PV-and-BU-only setups or traditional load distribution strategies.

Electric vehicles represent a growing category of electrical consumers, whose presence within the distribution network is anticipated to rise significantly in the near future. For this reason, planning and incorporating electric vehicles into the system has been identified as an area for further development. Demand response serves as a highly efficient energy management strategy, capable of enhancing both the technical performance and economic viability of the network. This approach has similarly been earmarked for exploration in future initiatives. Virtual Power Plants, positioned at points of consumption, offer distinct reliability benefits. In the event of a network fault, VPPs can supply energy to substantial portions of consumers, thus contributing to improved network reliability, a focus designated for future work in the proposed framework. Another critical metric within the distribution network is voltage security. By managing their active power output, VPPs have the potential to play a pivotal role in bolstering voltage security. This aspect has been prioritized for evaluation in forthcoming efforts under the proposed plan.