A novel distributed approach for event-triggered economic dispatch of energy hubs under ramp-rate limits integrated with sustainable energy networks

Energy Reports 10 (2023) 4097–4111 Available online 31 October 2023 2352-4847/© 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Energy Reports journal homepage: www.elsevier.com/locate/egyr Research paper A novel distributed approach for event-triggered economic dispatch of energy hubs under ramp-rate limits integrated with sustainable energy networks Ijaz Ahmed a,∗, Muhammad Rehan a, Abdul Basit a, Muhammad Tufail a, Nasim Ullah b, Marian Piecha c, Vojtech Blazek d, Lukas Prokop d aDepartment of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Islamabad, Pakistan bDepartment of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia cMinistry of Industry and Trade, Prague 11015, Czech Republic dENET Centre, VSB—Technical University of Ostrava, 708 00 Ostrava, Czech Republic ARTICLE INFO Keywords: Distributed optimization Energy networks Consensus control RETScreen® Energy sustainability Energy dispatch problem Event-triggered communication networks Hybrid energy systems networks ABSTRACT This paper investigates a new consensus-oriented distributed approach for the event-triggered (ET) economic dispatch problem (EDP) over a smart grid under ramp-rate limits (RRLs) integrated with green power sources (GPSs) such as solar and wind energy for demand response strategies over hybrid energy power systems. To address the RRL condition, the authors have transformed the RRLs as minimum and maximum bounds on the derivative of generation for a generator. Then, a Karush–Kuhn–Tucker (KKT) condition and a more practical approximate KKT condition are developed for determining the optimality conditions. A practical ET protocol is proposed over a topology between generators by application of the proposed approximate KKT condition. In contrast to existing distributed optimization methods, this paper provides both optimally condition and distributed optimization scheme for dealing with RRLs integrated with sustainable hybrid energy systems. In addition, a computationally efficient ET mechanism, eliminating Zeno behaviour, has been considered for dealing with the efficient utilization of communication resources. This study incorporates the real-time input data from thermal production plants and GPSs for experimental analysis. RETScreen software having data-set of over 6,700 local meteorological stations is applied to obtain input data for GPSs. Furthermore, Weibull and Beta distribution functions have been applied for dealing with uncertainty in wind and solar energy sources. Two case studies are examined (with and without GPSs), and simulation results demonstrated the suitable performance of the proposed distributed ET EDP approach. 1. Introduction Economic dispatch problem (EDP) is one of the most important concerns in smart grids, the aim of which is to address the power distribution in a network by minimizing the operating costs of power units by incorporating the system constraints (He et al.,2018;Bai et al., 2019;Ahmed et al.,2023c,2022b). Numerous methodologies have considered the centralized EDP through dynamic programming (Wang et al.,2020), gradient method (Massaroli et al.,2022), and Raphson iterative method (Murphy and Niebur,2021). In all these approaches, a global control centre is required to compute the operational cost and to obtain generation output of a generation node. These approaches require a central dispatch facility, which can be prone of system failure ∗Corresponding author. E-mail addresses: [email protected] (I. Ahmed), [email protected] (M. Rehan), [email protected] (A. Basit), [email protected] (M. Tufail), [email protected] (N. Ullah), [email protected] (M. Piecha), [email protected] (V. Blazek), [email protected] (L. Prokop). due to a fault in the dispatch centre. When a power system is partitioned into several subsystems, then a subsystem can apply the dynamic programming locally without using dispatch centre, as observed in Carli and Dotoli (2020). Modern computation technologies have facilitated the distributed processes, based on learning and training of models, for each generating unit by allowing fast adjustment of the power outputs and by accounting a scalable optimization dilemma. The performance of a distributed gradient-based method depends on computation of gradient and updation of state for each unit for eliminating the need of a central computation facility (Peres et al.,2018;Ahmed et al., 2023a). Similar approach can also be achieved for Newton–Raphson iterations without a central control facility, as seen in Irving and Sterling (1987). Designing a distributed EDP method, without using a https://doi.org/10.1016/j.egyr.2023.10.078 Received 28 March 2023; Received in revised form 20 October 2023; Accepted 25 October 2023 Energy Reports 10 (2023) 4097–4111 4098 I. Ahmed et al. Nomenclature 𝑋and 𝜛Distribution parameters 𝑎𝑖,𝑏𝑖and 𝑐𝑖Input fuel coefficients 𝐵𝑖𝑔 Loss coefficient matrix 𝐹𝑇and 𝑃𝑡Thermal cost and output power generation 𝐿Laplacian matrix 𝑀Amount of radiation 𝑁Number of units 𝑛and ℏTurbine shape and scale constants 𝑃𝐷and 𝑃𝐿Active demand power and power losses 𝑃min 𝑖and 𝑃max 𝑖Minimum and maximum active powers 𝑃𝑠and 𝑃𝑤Generated output power of solar and wind 𝑣Wind velocity 𝑣𝑖∕𝑝,𝑣𝑅, and 𝑣𝑜∕𝑝Input wind velocity, rated wind velocity and output wind velocity 𝐴, 𝐷, Adjacency and degree matrices 𝐷𝑅𝑖and 𝑈𝑅𝑖Upward and downward RRLs Acronyms ET Event-triggered ED Economic dispatch EDP Economic dispatch problem RRLs Ramp-rate limits GPSs Green power sources KKT Karush–Kuhn–Tucker IC Incremental cost PDF Probability density function NASA National Aeronautics and Space Administration central facility, can be advantageous, but it can be complicated to solve such a constrained optimization problem (Akram et al.,2018;Savkin et al.,2016;Khalid et al.,2018;Salman et al.,2020;Alzahrani et al., 2019;Savkin et al.,2014;Abdulgalil et al.,2019;Rauf et al.,2021;Abd El-Aziz et al.,2016). When compared to the typical centralized techniques (Zhou et al., 2015;Ahmed et al.,2022c;Alvi et al.,2022;Ahmed et al.,2022e), the distributed approaches have several significant advantages. Distributed techniques are more cost effective in terms of processing and communication in large-scale energy hubs (Li et al.,2018;Ahmed et al., 2022f;Li et al.,2012). Moreover, they are more versatile and reliable, making it more suited for smart grid applications (Liu and Song, 2022). Using a localized energy internet paradigm, the study (Hua et al.,2023) investigated the relationship among loss of energy and emission of carbon dioxide. The study developed a novel approach for energy allocation by combining the U-NSGA-III with DFS to identify feasible arrangement of sources and pathways. Nevertheless, another important feature of the distributed techniques is their sustainability in terms of clean energy, as they facilitate the rapid adoption of sustainable distributed generation equipment such as solar and wind systems (Li et al.,2022). Furthermore, distributed approaches support plug-and-play, which is a fundamental requirement of advanced smart grids; because centralized techniques necessitate an extra framework to accommodate the addition or elimination of generating units for maintenance purposes (Nudell et al.,2022). The advancement in the consensus algorithms has led the scientists to use distributed protocols to resolve the EDP concerns (Taousser et al.,2016). The major advantage of such protocols is the use of local information, rather than the global one. In recent years, there has been an increase of research studies, conducted on the distributed optimization and consensus control (see, e.g., Ahmed et al. (2022g), Gao et al. (2021), Wakaiki et al. (2021), Sun et al. (2022) and Rehan et al. (2017,2019), Basit et al. (2023b), Razaq et al. (2019), Basit et al. (2022a), Karaki and Mahmoud (2021) as well as the references contained therein). Using distributed economic dispatch, Xu et al. (2021a) reduced the overall cost of integrated micro-grid engaged in frequency control by taking into account the dynamical properties of inverter-based grids. Usually, generating systems are geographically distributed, feeding many feeders complicates the power delivery operation. To elude this limitation, the authors in Yin and Sun (2021) suggested a multi-layer distributed consensus algorithm for a largescale multi-zonal dispatch, connected through tie-lines in a network topology, to attain optimal power for each zone. However, handling machine-dependent constraints like ramp-rate limits (RRLs) are fundamental challenges in the distributed EDP operation and cannot be dealt through the above-presented approaches. The consideration of ramprate constraint is vital for generators, as it improves system security against excessive power flow rates and is needed for a sustainable operation. There has been an increasing interest of researchers worldwide for considering green power sources (GPSs), including wind and solar power plants for the recent years. EDP with renewable systems has been particularly considered owing to widespread use of GPSs in futuristic energy grids. The consideration of EDP methods by incorporating the renewable power predictions has been studied in Wang et al. (2021a). Because of uncertain fluctuations in solar irradiance and wind velocity, the production of energy from the wind turbines and solar plants is highly unpredictable. As a result, new strategies are being employed to cope with the uncertain behaviour of GPSs in the EDP dilemma. Consequently, power production through a wind power plant was reevaluated using a two-stage paradigm (Li et al.,2015a) through backup generators. Authors in Güvenç and Kaymaz (2019) have employed a coyote optimization algorithm to solve the EDP, integrated with GPSs. Liao et al. proposed a chaotic quantum genetic algorithm to resolve the emission reduction in hybrid power systems (Liao,2011). The works in Ahmed et al. (2022e,d) have proposed heuristic-based solutions to attain the optimal generation cost by integrating the traditional EDP model with solar energy. Point-estimation approaches have been employed to mitigate the uncertainty of the solar radiation. A consensus driven protocol for economic dispatch model considering solar energy was presented in Moin et al. (2022). GPSs have been increasingly adapted along with the conventional power grids as part of a global trend towards sustainability (Aziz et al.,2023). In this context, a stochastic optimization approach has been applied to resolve the EDP for optimal management, and a bench mark test system (IEEE 14 bus system) has been utilized to evaluate and validate the resultant method. Several other studies have demonstrated efficacy to resolve the complex EDP (Zhang et al.,2022a;Deng et al.,2022;Cui et al.,2023;Dashtdar et al.,2022); however, these methods are computationally complex to implement as well as demand extra data processing capabilities and network bandwidth. Furthermore, most of these methods are vulnerable to cyber-attacks and faults owing to the centralized nature of dispatch facility. There will be a significant amount of data transmission for EDP operation in a smart grid, putting a stress on communication networks (Gungor et al.,2011;Zou et al.,2021). On the other hand, distributed techniques appear to be a potential method for resolving the complicated EDP with GPSs integration. The event-triggered (ET) schemes have been proposed and tested using various control mechanisms and communication topologies. These approaches aim to reduce the number of samples taken, consequently, conserving the computational resources (Basit et al.,2022c;Zhang et al.,2022b;Basit et al.,2022b,2023c). The work of Li et al. (2015) has developed the 𝜃-logarithmic approach to re-design the EDP approach via an event-triggering mechanism. Ying et al. have developed a distributed algorithm for attaining the EDP by means of ET mechanism on a network of uncertain communication topologies (Wan et al.,2021). The Energy Reports 10 (2023) 4097–4111 4099 I. Ahmed et al. authors of the study (Dong et al.,2023) has introduced an algorithm for achieving the convergence to the optimal level via a linear convergence rate. This desired convergence performance has been achieved through appropriate selection of operational parameters under convex objective functions and uniform triggering. A through examination of the ET EDP can be found in the works (Yuan et al.,2023;Xu et al.,2021b). The previous literature and the works like Jin et al. (2020) have significantly increased the understanding of EDP operation in smart grids under diverse scenarios. However, these approaches do not address the EDP fundamental operational constraints, such as ramp-rate conditions. Further, the consideration of ET design for obtaining a better solution in terms of communication bandwidth is lacking. Distributed optimization is relatively a new dilemma in research, which needs more attention for handling the system-level and communication-level constraints from both control and network perspectives. These research gaps in the existing works form the primary motivation of the present study. Based on the mentioned research gaps, this paper addresses an ET consensus-oriented approach for the distributed EDP of power generators, integrated with GPSs, over a network of smart grid. Additionally, the issue of ramp-rate constraints has also been accounted for a matterof-fact investigation. First, conditions for the optimization of total cost function (summation of individual costs) under RRLs are investigated through Lagrange optimization. We have also provided an approximate Karush–Kuhn–Tucker (KKT) condition to deal with practical ET issues. The provided approximate KKT condition ensures a balance of power for supply–demand, while handling the system-confined RRL condition in a dynamic fashion. Then, a nonlinear ET protocol for dealing with RRLs has been achieved. The convergence of this nonlinear approach has been proved through Lyapunov theory and the newly developed optimality condition. The proposed method offers numerous benefits. It facilitates efficient management of the dispatch operation by computing the optimal incremental costs (ICs) for all the units concerned. This optimization approach has been attained through the proposed optimality condition, which uses the local data for precise decision-making. To the best of our knowledge, an ET distributed EDP method under RRLs along with the convergence analysis has been addressed for the first time. By taking into account local characteristics and constraints, the proposed method enables a precise and cost-effective allocation of power resources. Consequently, it enhances the system performance and provides the cost effectiveness. The key developments of this study can be summarized as follows: (1) the introduction of a nonlinear protocol to handle RRLs, (2) a reliance on local information for decision-making, (3) the inclusion of a virtual state for monitoring purposes, and (4) an emphasis on the ET mechanism rather than time-triggered approaches. Through these advancements, this work contributes to the field by addressing the research gaps and by providing a novel solution for an efficient and an effective management of distributed EDPs under RRLs. In comparison to the closely related work (He et al.,2018), there are several significant advancements offered by the proposed work. First, the suggested approach introduces a nonlinear protocol specifically designed to effectively handle the challenges posed by RRLs. This quality distinguishes the proposed method from the previous ones that rely on the conservative linear protocols. Second, this work applies the local information from the adjacent nodes, leveraging it as a crucial input for the decision-making systems (Ahmed et al.,2022c,e). Thirdly, this work incorporates the concept of a virtual state, which can be utilized as a monitoring mechanism within the generating system. The main contributions of this work are as follows: 1. A KKT condition under RRL constraint has been devised by transforming the RRL constraint into generation derivative bounds and by analysing the Lagrange approach. This fundamental KKT condition can be helpful in dealing with the distributed EDP integrated with GPSs under RRLs. Its application within distributed ET optimization domain provides a reliable tool for effectively managing and optimizing the distributed EDP-GPS network under the constraints imposed by RRLs. 2. In contrast to the works in Ahmed et al. (2023d,2014,2022e), the proposed approach quickly converges to the optimal solution with less computational time and meets the desired criteria while effectively handling the time constraints such as RRLs. Compared with approaches in Bai et al. (2019), Xu et al. (2021a), Wan et al. (2021), Wang et al. (2021b) and Liu et al. (2018), the suggested protocols ensure the Lyapunov stability criteria and all agents reach on practical consensus within finer instance of time. 3. Further, a new 𝜀𝑖-KKT condition is introduced as an approximation of the proposed KKT condition, as seen in the fundamental details in Dutta et al. (2013). This 𝜀𝑖-KKT condition offers a viable approach for addressing the EDP and for achieving a specified tolerance level, particularly, in the presence of an ET mechanism. Notably, this 𝜀𝑖-KKT condition presents a distinct advantage of tolerance, compared to the existing condition in He et al. (2018). 4. The conventional distributed EDP protocols, such as those presented in Bai et al. (2019), Gao et al. (2021), Xu et al. (2021a), Li et al. (2015), Wan et al. (2021), Wang et al. (2021b) and Liu et al. (2018), have been modified to incorporate an additional nonlinear dynamic component to effectively address the RRLs. The proposed approach involves the implementation of virtual IC consensus instead of actual IC consensus, attained through the introduction of an additional nonlinear dynamical mechanism. This additional dynamical element ensures the accurate tracking of RRLs by enforcing the saturation of the generator’s generation rate. The suggested modifications have been carefully designed in order to manage the challenges posed by RRLs without compromising the optimal operation in a smart grid network. 5. The convergence analysis of the proposed approach, achieved via the consideration of RRLs and the attainment of supply– demand balance, has been rigorously established by application of the Lyapunov stability theory. This intricate analysis has been conducted through a six-step approach in order to achieve an optimal generation cost for all the committed generation units by managing the 𝜀𝑖-KKT condition. It is worth noting that the convergence analysis in the present study is quite challenging, in contrast with existing methods (Bai et al.,2019;Gao et al.,2021; Xu et al.,2021a;Li et al.,2015;Wan et al.,2021;Wang et al., 2021b;Liu et al.,2018), due to the presence of nonlinearity (required for maintaining RRLs), the incorporation of an additional virtual state (for system monitoring purposes), and the integration of the ET mechanism (for efficient communication resource utilization). 6. The proposed approach adopts an ET mechanism that facilitates fewer communication resources among generators by application of the local information. The elimination of the Zeno behaviour, which poses practical challenges, has also been achieved in the proposed method. This achievement sets the suggested approach apart from previous works such as Bai et al. (2019), Xu et al. (2021a), Wang et al. (2021b) and Liu et al. (2018) for communication bandwidth. In comparison to the existing ET approach presented in Wan et al. (2021), the suggested method introduces a computationally efficient ET mechanism. This computational simplicity has been attributed for the utilization of the 𝜀𝑖-KKT condition, which has improved the ET process. 7. In contrast to the previous studies (Bai et al.,2019;Gao et al., 2021;Xu et al.,2021a;Li et al.,2015;Wan et al.,2021;Wang et al.,2021b;Liu et al.,2018), the proposed EDP model incorporates the integration of GPSs. To obtain improved power predictions, the RETScreen software has been employed by considering the real-time dataset, having approximately 6700 local meteorological stations. This GPS input data and the RETScreen software utilization ensure reliable and practical prediction of power generation by application of the proposed EDP model. Energy Reports 10 (2023) 4097–4111 4100 I. Ahmed et al. 2. System description 2.1. Graph theory A typical power system has multi-way communication, which can be represented by an undirected graph 𝐺=(𝑉 , 𝐸), where 𝑉= {𝑣1, 𝑣2,…, 𝑣𝑁}is the set of graph nodes that denotes 𝑁generators in the power system. 𝑉×𝑉has subset 𝐸= {𝑒𝑙= {𝑣𝑖, 𝑣𝑗}|||𝑣𝑖, 𝑣𝑗∈𝑉}. The two connected vertices are referred to as neighbours, since they are close to one another. The graph’s adjacency, degree, and graph Laplacian matrices are specified as 𝐴, 𝐷, and 𝐿, respectively. The graph’s adjacency relationships are represented by the 𝑁×𝑁symmetric adjacency matrix 𝐴=[𝛼ij]. The degree matrix 𝐷is a diagonal matrix with each diagonal element representing the in-degree of the corresponding node. A vital factor in graph theory is the Laplacian 𝐿, which relates to the graph dynamic properties and is defined as 𝐿=𝐷−𝐴(Basit et al.,2023a). 2.2. Distributed wind-solar ED modelling under RRL Consider the cost function of thermal generating units as (Ahmed et al.,2023d) 𝐶𝑖(𝑃𝑖)=𝑎𝑖𝑃2 𝑖+𝑏𝑖𝑃𝑖+𝑐𝑖,∀𝑖= 1,2,…, 𝑁. (1) The cost function of wind and solar generation can be expressed as (Ahmed et al.,2022d) 𝐹𝑠(𝑃𝑠,𝑚)= 𝑁𝑠 ∑ 𝑚=1 𝑃𝑠,𝑚𝑧𝑖𝑑𝑚,(2) where in expression (2),𝑃𝑠,𝑚 represents the cumulative solar power production in MW, 𝑧𝑖𝑑𝑚signifies the pricing rate for the 𝑚th solar DG in ($/MW), and the summation ∑𝑁𝑠 𝑚=1 𝑃𝑠,𝑚𝑧𝑖𝑑𝑚corresponds to the overall cost of electricity generation from all active photovoltaic facilities. 𝐹𝑤(𝑃𝑤,𝑛)= 𝑁𝑤 ∑ 𝑛=1 𝑃𝑤,𝑛𝐾𝑊𝑛.(3) The total cost function of wind-solar ED can be expressed as (Ahmed et al.,2022d) 𝐶𝑖(𝑃𝑖, 𝑃𝑠,𝑚, 𝑃𝑤,𝑛) = 𝐹𝑇(𝑃𝑡) + 𝐹𝑤(𝑃𝑤,𝑛) + 𝐹𝑠(𝑃𝑠,𝑚).(4) The quadratic function in (1) represents the generation cost of 𝑖th unit. Here 𝑎𝑖,𝑏𝑖and 𝑐𝑖are the input fuel coefficients of 𝑖th unit. 𝐹𝑇and 𝑃𝑡are the thermal cost and output power generation of 𝑖th committed unit, respectively. The cost function in (2) and (3) represent the wind-solar generation cost which are usually fixed (installation and operational cost) and taken as constant. 𝑃𝑠and 𝑃𝑤represents the generated output power of solar and wind, respectively. The fluctuation in wind speed is the most important factor in wind energy. Several methods have been developed to characterize the unpredictable nature of wind velocity patterns. In this work, the features of wind velocity are modelled using the Weibull Probability Density Function. Nonlinear features of wind velocity profiles are typically described using the Weibull Probability Density Function approach. This distribution function (𝐹𝑝𝑑𝑓 )can be written as (Ahmed et al.,2022d) 𝑓𝑝𝑑𝑓 (𝑣) = 𝑛 ℏ(𝑣 ℏ)𝑛−1 exp (−(𝑣 ℏ)𝑛)(𝑣 > 0).(5) In expression (5),𝑛and ℏdenote the turbine shape and scale constants, 𝑣denote the wind velocity, 𝑣𝑖∕𝑝,𝑣𝑅, and 𝑣𝑜∕𝑝denote rated, cut-in, and cut-out velocities respectively. Using a mapping from wind velocity to electrical output, the wind generators energy generation can be characterized as a stochastic process assuming the wind’s unpredictability. By employing the velocity model, one can determine the amount of energy that can be harnessed from the wind based on a specific wind velocity. 𝑃𝑤,𝑛 =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0, 𝜐 < 𝜐𝑖∕𝑝𝑜𝑟 𝜐 > 𝜐𝑜∕𝑝, 𝑄𝑅(𝜐−𝜐𝑖∕𝑝) 𝜐𝑅−𝜐𝑖∕𝑝(𝜐𝑖∕𝑝⩽𝜐⩽𝜐𝑅), 𝑄𝑅(𝜐𝑅⩽𝜐⩽𝜐𝑜∕𝑝). (6) Solar panels capacity to produce electricity is determined by elements such as solar irradiation, weather conditions, and the panels design. In order to replicate solar radiation, the beta PDF model is utilized (Ahmed et al.,2022e). 𝐹𝛽(𝑀)=⎧ ⎪ ⎨ ⎪ ⎩ 𝐸(𝑋+𝜛) 𝐸(𝑋)𝐸(𝜛)×𝑀𝑋−1(1 − 𝑀)𝜛−1, 𝑓𝑜𝑟 0⩽𝑀⩽1, 𝑋 ⩾0, 𝜛 ⩾0, 0𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (7) In expression (7) 𝑋and 𝜛are distribution parameters, 𝑀is amount of radiation panels received and 𝐸represents the PDF distribution function. The estimation of 𝑋and 𝜛can be stated as follow. 𝑋=𝐵(𝐵(𝐵+ 1) 𝜙2− 1),(8) 𝜛=(1 − 𝐵)((𝐵(𝐵+ 1) 𝜙2− 1)).(9) In expressions (8) and (9),𝜙represents the standard deviation, providing a measure of the dispersion or spread of the data points around the mean 𝐵. As mentioned earlier, radiation from the sun and weather conditions have a major impact on solar production, which can be described in the following way (Ahmed et al.,2022d). 𝑃𝑠,𝑚(𝑡) = 𝑁𝑠,𝑆𝑅 ×𝑁𝑃 𝐿[𝑃𝑠(𝑆𝐶)× 𝑃(𝑡)𝑅𝐷 𝑃𝑅𝐷.𝑆𝐶 ×[1−𝛩×(𝐻𝐶𝑒𝑙 −𝐻𝐶𝑒𝑙.𝑆𝐶 )]],(10) 𝐻𝐶𝑒𝑙 =𝑃𝐴𝑇 +𝑃(𝑡)𝑅𝐷 𝑃𝑅𝐷.𝑆𝐶 × (𝑃𝑁𝑇 − 20).(11) Now suppose there are 𝑁thermal units participating in a grid and the goal of the EDP is to plan the power outputs for the operating units in such a way that the overall power system operates at optimal grid cost by satisfying the associated system limitations. As a result, an optimization problem can be specified as 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 ∑𝑁 𝑖=1𝐶𝑖(𝑃𝑖(𝑡))+𝐹𝑠(𝑃𝑠,𝑚) + 𝐹𝑤(𝑃𝑤,𝑛), 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ∑𝑁 𝑖=1𝑃𝑖(𝑡) + 𝑃𝑠,𝑚 +𝑃𝑤,𝑛 =𝑃𝐷(𝑡) + 𝑃𝐿(𝑡), 𝑃min 𝑖≤𝑃𝑖≤𝑃max 𝑖, −𝐷𝑅𝑖≤𝑃𝑖(𝑡)−𝑃𝑖(𝑡−𝜏)≤𝑈𝑅𝑖, (12) where 𝑃min 𝑖and 𝑃max 𝑖are the minimum and maximum active power limits of 𝑖th unit for all 𝑖= 1,2,…, 𝑁.𝑃𝐷and 𝑃𝐿are the active demand power and power losses, respectively. The 𝐷𝑅𝑖and 𝑈𝑅𝑖represent the downward and upward ramp limits of 𝑖th unit. Power losses in a network are represented by 𝑃𝐿with loss coefficient matrix 𝐵𝑖𝑔, and its relation is given by (Ahmed et al.,2022e): 𝑃𝐿= 𝑁 ∑ 𝑖=1 𝑁 ∑ 𝑔=1 𝑃𝑖𝑃𝑔𝐵𝑖𝑔.(13) Remark 1. Many prior distributed EDP methods like Bai et al. (2019), Gao et al. (2021), Xu et al. (2021a), Li et al. (2015), Wan et al. (2021), Wang et al. (2021b) and Liu et al. (2018) have not accounted the RRLs. The presented EDP model of (12) takes into account the RRL constraints, as thermal unit can only vary their output powers with respect to the previous values by a certain amount practically. The focus of the proposed study is to assist in regulating the change of power of committed units for the distributed optimization in contrast to the existing distributed schemes. Initially, the total generation of 𝑖th unit must be capable to meet the network power demand and power losses in the transmission system. The following assumption has been considered to put forth the scenario. Energy Reports 10 (2023) 4097–4111 4101 I. Ahmed et al. Assumption 1. Suppose that the initial generation validates the following expression: 𝑁 ∑ 𝑖=1 𝑃𝑖(0) = 𝑃𝐷+𝑃𝐿−𝑃𝑠,𝑚 −𝑃𝑤,𝑛.(14) Assumption 2. The graph 𝐺=(𝑉 , 𝐸)between generators is strongly connected. The strong connectivity assumption for consideration of a directed graph requires the existence of a directed path between every node pair, leading to an unrestricted connectivity. This property has been used in a variety of disciplines, like distributed systems, network design, and routing algorithms. Strong connectivity is needed in the context of consensus and optimization problems. The consensus and optimization problems aim to achieve coordination through sharing of information over a network of connected agents (in the present case, generators). Strong connectivity ensures an efficient exchange of signals between agents, consequently, facilitating the convergence of the algorithm towards an optimal point. A sub-problem of the proposed problem (12) is reformulated and further investigated by excluding the capacity limit, which will be addressed later. The resultant problem becomes 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 ∑𝑁 𝑖=1 𝐶𝑖(𝑃𝑖(𝑡))+𝐹𝑠(𝑃𝑠,𝑚) + 𝐹𝑤(𝑃𝑤,𝑛), 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ∑𝑁 𝑖=1 𝑃𝑖(𝑡) + 𝑃𝑠,𝑚 +𝑃𝑤,𝑛 =𝑃𝐷+𝑃𝐿, −𝐷𝑅𝑖≤𝑃𝑖(𝑡)−𝑃𝑖(𝑡−𝜏)≤𝑈𝑅𝑖, 𝑖 = 1,…, 𝑁. (15) The ramp-rate constraints stated in (12) make the problem more difficult to deal with because these conditions include the difference between generations at two different time instants. The problem at hand will be addressed by employing the generation rate, which involves partitioning the final condition by 𝜏to achieve −𝐷𝑅𝑃 𝑖≤𝑃𝑖(𝑡) − 𝑃𝑖(𝑡−𝜏) 𝜏≤𝑈𝑅𝑃 𝑖, where 𝑈𝑅𝑃 𝑖=𝑈𝑅𝑖∕𝜏and 𝐷𝑅𝑃 𝑖=𝐷𝑅𝑖∕𝜏. It further reveals that −𝐷𝑅𝑃 𝑖≤𝑑𝑃𝑖(𝑡) 𝑑𝑡 ≤𝑈𝑅𝑃 𝑖.(16) Therefore, an equivalent problem of (15) is written as 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑁 ∑ 𝑖=1 𝐶𝑖(𝑃𝑖(𝑡))+𝐹𝑠(𝑃𝑠,𝑚) + 𝐹𝑤(𝑃𝑤,𝑛), 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑁 ∑ 𝑖=1 𝑃𝑖(𝑡) + 𝑃𝑠,𝑚 +𝑃𝑤,𝑛 =𝑃𝐷+𝑃𝐿, (17) −𝐷𝑅𝑃 𝑖≤𝑑𝑃𝑖(𝑡) 𝑑𝑡 ≤𝑈𝑅𝑃 𝑖, 𝑖 = 1,…, 𝑁. The aims of the present study are (i) to investigate the optimality conditions for the distributed optimization problem (15) or (17), (ii) to determine an appropriate consensus-based optimization protocol for the optimization of (15), (iii) to investigate stability analysis of the proposed algorithm, and (iv) to consider ET mechanism for solving the mentioned problem. 3. Proposed optimality conditions This section will elucidate the optimality conditions associated with the EDP considering the constraints imposed by ramp rates. For problem in (15) or (17), two conditions are presented known as KKT and relaxed 𝜀𝑖-KKT conditions, as seen in Dutta et al. (2013) by the application of Lagrange function approach (Sabermahani et al.,2021; Ito and Kunisch,2008). A KKT condition for solving (17) is presented as follows. Lemma 1. An optimal solution 𝑃∗ 𝑖to the problem (17) is obtained, if the following conditions are satisfied: 𝜕𝐶𝑖(𝑃𝑖) 𝜕𝑃𝑖 =𝜕𝐶𝑗(𝑃𝑗) 𝜕𝑃𝑗 ,(18) 𝑁 ∑ 𝑖=1 𝑃𝑖+𝑃𝑠,𝑚 +𝑃𝑤,𝑛 =𝑃𝐷+𝑃𝐿,(19) −𝐷𝑅𝑃 𝑖≤ 𝑃𝑖(𝑡)≤𝑈𝑅𝑃 𝑖.(20) Proof. Consider a Lagrange function L(𝑃𝑖)for (17) as L(𝑃𝑖) = 𝑁 ∑ 𝑖=1 𝐶𝑖(𝑃𝑖) + 𝜆[𝑃𝐷+𝑃𝐿− 𝑁 ∑ 𝑖=1 𝑃𝑖−𝑃𝑠,𝑚 −𝑃𝑤,𝑛] + 𝑁 ∑ 𝑖=1 𝜆𝑈 𝑖(𝑈𝑅𝑃 𝑖− 𝑃𝑖−𝜒𝑈 𝑖) + 𝑁 ∑ 𝑖=1 𝜆𝐷 𝑖( 𝑃𝑖+𝐷𝑅𝑃 𝑖−𝜒𝐷 𝑖), (21) where 𝜆, 𝜆𝑈 𝑖, 𝜆𝐷 𝑖for 𝑖= 1,2,…, 𝑁 are Lagrange multipliers and 𝜒𝑈 𝑖≥0 and 𝜒𝐷 𝑖≥0are free variables. For optimal solution, we need 𝜕L(𝑃𝑖) 𝜕𝑃𝑖 = 0,(22) 𝜕L(𝑃𝑖) 𝜕𝜆 = 0,(23) 𝜕L(𝑃𝑖) 𝜕𝜆𝑈 𝑖 = 0,(24) 𝜕L(𝑃𝑖) 𝜕𝜆𝐷 𝑖 = 0.(25) As 𝜕 𝑃𝑖(𝑡) 𝜕𝑃𝑖 = 0,(22) and (23) reveal (18) and (19), respectively. The conditions (24) and (25) lead to 𝑈𝑅𝑃 𝑖− 𝑃𝑖=𝜒𝑈 𝑖≥0,  𝑃𝑖+𝐷𝑅𝑃 𝑖=𝜒𝐷 𝑖≥0. These conditions further lead to 𝑈𝑅𝑃 𝑖≥ 𝑃𝑖(𝑡)and  𝑃𝑖(𝑡)≥−𝐷𝑅𝑃 𝑖, respectively, which are equivalent to (20), and it complete the proof. ■ Remark 2. The RRL has been considered in the optimality condition of Lemma 1 by application of Lagrange function (21) and by introducing the free variables 𝜒𝑈 𝑖≥0and 𝜒𝐷 𝑖≥0. It is important to note that Lemma 1 is an effective tool for achieving a protocol for realizing a distributed optimum solution 𝑃∗ 𝑖to the optimization problem (17). The proposed condition meets with the primary constraint of generation while simultaneously fulfilling the ramp restriction. The condition has been derived by converting the ramp-rate constraint −𝐷𝑅𝑖≤𝑃𝑖(𝑡)−𝑃𝑖(𝑡−𝜏)≤𝑈𝑅𝑖into generation derivative constraint −𝐷𝑅𝑃 𝑖≤𝑃𝑖(𝑡)−𝑃𝑖(𝑡−𝜏) 𝜏≤𝑈𝑅𝑃 𝑖, the later one is easier to realize and can be handled dynamically via an optimization protocol. Although KKT conditions can be attained at the optimal point, but the conventional KKT condition has inherent restrictions, and it cannot be utilized to directly measure the performance of an optimization approach. Consequently, alternative methods are available, which can address the tolerance of the solution. Here we adapt the following definition: Definition 1. An approximate solution to the problem (17) is said to be satisfy the 𝜀𝑖-KKT condition (Dutta et al.,2013), if the constraint |||| 𝜕L(𝑃𝑖) 𝜕𝑃𝑖||||≤𝜀𝑖(26) is satisfied along with the conditions (23),(24) and (25). Energy Reports 10 (2023) 4097–4111 4102 I. Ahmed et al. Based on Definition 1, the previous result of Lemma 1 is modified in Lemma 2. A relaxed 𝜀𝑖-KKT condition for solving (17) is presented to attain a solution in the neighbourhood of the optimal solution 𝑃∗ 𝑖, and is given as follows. Lemma 2. An 𝜀𝑖-KKT solution to the problem (17) is obtained, if for a positive scalar 𝜉the conditions (19),(20) and ||||| 𝜕𝐶𝑖(𝑃𝑖) 𝜕𝑃𝑖 −𝜕𝐶𝑗(𝑃𝑗) 𝜕𝑃𝑗||||| ≤𝜉(27) are satisfied for 𝑖, 𝑗 = 1,2,…, 𝑁, 𝑖 ≠𝑗. Proof. As seen in the proof of Lemma 1, the relations (23)–(25) lead to (19)–(20). The condition (26) along with (21) leads to |||| 𝜕𝐶𝑖(𝑃𝑖) 𝜕𝑃𝑖 −𝜆||||≤𝜀𝑖.(28) Hence, one can determine the range of the IC as 𝜕𝐶𝑖(𝑃𝑖) 𝜕𝑃𝑖 ∈[𝜆−𝜀𝑖, 𝜆 +𝜀𝑖]. It further reveals that the difference between two ICs of two different generators belong the range, given by 𝜕𝐶𝑖(𝑃𝑖) 𝜕𝑃𝑖 −𝜕𝐶𝑗(𝑃𝑗) 𝜕𝑃𝑗 ∈[−𝜀𝑖−𝜀𝑗, 𝜀𝑖+𝜀𝑖], for 𝑖, 𝑗 = 1,2,…, 𝑁 and 𝑖≠𝑗. Consequently, (27) is validated for 𝜉= 2,max𝑖=1,2,…,𝑁 𝜀𝑖, which ends the proof. ■ Remark 3. In comparison to the condition in Lemma 1, the result of Lemma 2 has a number of advantages. First, it can provide a term for setting the tolerance value 𝜉of the solution (more specifically a stopping criterion). Second, because the authors are dealing with an ET scenario, the resultant scheme from Lemma 2 can be useful in terms of offering an approximate solution with respect to the triggering error. Additionally, it can also cope with the numerical issues like round-off error, truncation error, and quantization error arising while implementation of protocols via digital processors. Remark 4. Compared to the recent investigation (He et al.,2018), the proposed optimality condition in Lemma 1 is more clear as it relates with the ICs and generation rates. Further, the condition in Lemma 2 is more interesting, as it can be applied to design a practical ET distributed optimization approach with consideration of tolerance level. 4. Proposed distributed ET EDP strategy Let us assign the IC variable as 𝜂𝑖=𝜕𝐶𝑖(𝑃𝑖) 𝜕𝑃𝑖 = 2𝑎𝑖𝑃𝑖(𝑡) + 𝑏𝑖, the proposed ET nonlinear second-order protocol for solving the optimization (17) has the form given as 𝑥𝑖(𝑡)=2𝑐𝑎𝑖 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑥𝑗(𝑡𝑗 𝑘) − 𝑥𝑖(𝑡𝑖 𝑘)),∀𝑖= 1,…, 𝑁, 𝜂𝑖=𝛹𝑅(𝜃(𝜂𝑖−𝑥𝑖) + 𝑥𝑖), 𝑥𝑖(0) = 𝜂𝑖(0),(29) 𝑃𝑖= (𝜂𝑖−𝑏𝑖)∕2𝑎𝑖, where 𝑥𝑖(𝑡)is the state of the protocol and can be referred as the virtual IC of 𝑖𝑡ℎ generator, 𝑥𝑖(𝑡𝑖 𝑘)is ET version of 𝑥𝑖(𝑡)at triggering instant 𝑡𝑖 𝑘, and 𝑐and 𝜃are protocol parameters, to be selected for the distributed optimization. A generator 𝑗shares the information of its triggered virtual IC with another generator 𝑖, which is applied for updating the virtual state of the generator 𝑖. Based on the updated virtual IC, each generator decides its real IC 𝜂𝑖through the second equation in the protocol. The function 𝛹𝑅(⋅)in the actual IC dynamics has a critical role in implementing RRLs. It can be selected via 𝛹𝑅(𝑢)=⎧ ⎪ ⎨ ⎪ ⎩ 2𝑎𝑖𝑈𝑅𝑃 𝑖,if𝑢≥2𝑎𝑖𝑈𝑅𝑃 𝑖, −2𝑎𝑖𝐷𝑅𝑃 𝑖,if𝑢≤−2𝑎𝑖𝐷𝑅𝑃 𝑖, 𝑢, otherwise, (30) for a scalar 𝑢. Remark 5. The proposed optimization protocol in (29) is novel due to consideration of 𝜂𝑖=𝛹𝑅(𝜃(𝜂𝑖−𝑥𝑖) + 𝑥𝑖), 𝑥𝑖(0) = 𝜂𝑖(0) and selection of 𝛹𝑅(⋅)via (30). These changes are employed for solving the distributed EDP under RRLs in contrast to the existing approaches (He et al.,2018; Bai et al.,2019;Gao et al.,2021;Xu et al.,2021a;Li et al.,2015;Wan et al.,2021;Wang et al.,2021b;Liu et al.,2018). The scenario is also different with existing methods, as the protocol is based on virtual state consensus, and the mentioned dynamics in (29) is employed to govern the RRLs. Let us define the triggering error as 𝑠𝑖(𝑡) = 𝑥𝑖(𝑡) − 𝑥𝑖(𝑡𝑖 𝑘). By application of (29), it yields 𝑥𝑖(𝑡)=2𝑐𝑎𝑖 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑥𝑗(𝑡) − 𝑥𝑖(𝑡)) +2𝑐𝑎𝑖 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑠𝑖(𝑡) − 𝑠𝑗(𝑡)). (31) The proposed event-triggering condition for the EDP is given as 𝑡𝑖 𝑘+1 = inf {𝑡 > 𝑡𝑖 𝑘|𝑠𝑇 𝑖(𝑡)𝑠𝑖(𝑡)≥𝜍𝑖}.(32) Let us define the consensus error between virtual ICs as 𝑒𝑖(𝑡) = 𝑥𝑖(𝑡)− 𝑥, for 𝑥 =𝑁−1 ∑𝑁 𝑗=1 𝑥𝑗(𝑡), the relation (31) leads to 𝑒𝑖(𝑡)=2𝑐𝑎𝑖 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑒𝑗(𝑡) − 𝑒𝑖(𝑡)) +2𝑐𝑎𝑖 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑠𝑖(𝑡) − 𝑠𝑗(𝑡)). (33) By applying the proposed 𝜀𝑖-KKT condition in Lemma 2, the following theorem establishes a distributed ET EDP solution under RRLs. Theorem 1. Consider 𝑁generators with cost functions in (1) satisfying Assumptions 1–2. There exists a local distributed ET 𝜀𝑖-KKT solution to the problem (17) via the algorithm (29) under 𝛹𝑅(⋅)in (30) through the eventtriggering condition (32) for scalars 𝑐and 𝜃of sufficiently small magnitudes, if 𝑐 > 𝜗∕2𝑞𝜆min(𝛬𝐿), 𝜃 < 0,(34) for positive scalars 𝑞and 𝜗, where 𝛬=𝑑𝑖𝑎𝑔{𝑎1, 𝑎2,…, 𝑎𝑁}. In addition, the proposed event-triggering optimization scheme eliminates the possibility of Zeno behaviour. Proof. Here, authors show that the proposed approach with eventtriggering condition ensures the 𝜀𝑖-KKT in Lemma 2. Additionally, the authors also demonstrate that the proposed event-triggering condition ensures elimination of Zeno behaviour. The proof has been provided in the six steps as follows: Step I: In the first step, it is demonstrated that the virtual ICs achieve a practical consensus, that is, the error 𝑒𝑖(𝑡)converges to a bounded region under the topology of Assumption 2. For this purpose, under scalar 𝑞 > 0, let us define a Lyapunov function as 𝑉(𝑒, 𝑡)=𝑞 𝑁 ∑ 𝑖=1 𝑒𝑇 𝑖(𝑡)𝑒𝑖(𝑡).(35) Energy Reports 10 (2023) 4097–4111 4103 I. Ahmed et al. Taking the time-derivative of (35) and substituting (33), it yields that  𝑉(𝑒, 𝑡)= 2𝑞𝑐 𝑁 ∑ 𝑖=1 𝑒𝑇 𝑖(𝑡)𝑎𝑖 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑒𝑗(𝑡) − 𝑒𝑖(𝑡)) +2𝑞𝑐 𝑁 ∑ 𝑖=1 𝑒𝑇 𝑖(𝑡)𝑎𝑖 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑠𝑖(𝑡) − 𝑠𝑗(𝑡)). (36) Applying 𝛬=𝑑𝑖𝑎𝑔{𝑎1, 𝑎2,…, 𝑎𝑁},𝑒𝑇(𝑡) = [𝑒𝑇 1⋯𝑒𝑇 𝑁]and 𝑠𝑇(𝑡) = [𝑠𝑇 1⋯𝑠𝑇 𝑁], we have  𝑉(𝑒, 𝑡)= −2𝑞𝑐𝑒𝑇(𝑡)𝛬𝐿𝑒(𝑡)+2𝑞𝑐𝑒𝑇(𝑡)𝛬𝐿𝑠(𝑡).(37) By employing the inequality 2𝑎𝑇𝑃 𝑏 ≤𝜀𝑎𝑇𝑃 𝑎 +𝜀−1𝑏𝑇𝑃 𝑏, the upper bound for the term in (37) for positive scalar 𝜗is obtained as 𝑒𝑇(𝑡)𝛬𝐿𝑠(𝑡)≤𝜗𝑞𝑒𝑇(𝑡)𝑒(𝑡) +𝜗−1𝑞𝑐2(𝜆max(𝛬𝐿))2∑𝑁 𝑖=1 𝑠𝑇 𝑖(𝑡)𝑠𝑖(𝑡).(38) Using (38) into (37), it yields  𝑉(𝑒, 𝑡)= − (2𝑞𝑐𝜆min(𝛬𝐿) − 𝜗)𝑞𝑒𝑇(𝑡)𝑒(𝑡) +𝜗−1𝑞𝑐2(𝜆max(𝛬𝐿))2 𝑁 ∑ 𝑖=1 𝑠𝑇 𝑖(𝑡)𝑠𝑖(𝑡).(39) Under 𝑠𝑇 𝑖(𝑡)𝑠𝑖(𝑡)< 𝜍𝑖and 𝜌=𝜗−1𝑞𝑐2(𝜆max(𝛬𝐿))2∑𝑁 𝑖=1 𝜍𝑖, we have  𝑉(𝑒, 𝑡)= − (2𝑞𝑐𝜆min(𝛬𝐿) − 𝜗)𝑞𝑒𝑇(𝑡)𝑒(𝑡) + 𝜌. (40) For the selection 𝑐 > 𝜗∕2𝑞𝜆min(𝛬𝐿), 𝑉(𝑒, 𝑡)<0under (2𝑞𝑐𝜆min(𝛬𝐿) − 𝜗) 𝑞𝑒𝑇(𝑡)𝑒(𝑡)> 𝜌, and the error will converge to the region 𝑒𝑇(𝑡)𝑒(𝑡)≤ 𝜌∕(2𝑞𝑐𝜆min(𝛬𝐿) − 𝜗)𝑞. Step II: In this step, it is demonstrated that the actual IC 𝜂𝑖(𝑡)can converge to the virtual IC 𝑥𝑖(𝑡)locally. Under the definition of 𝛹𝑅(⋅)in (30) along with the initial condition 𝑥𝑖(0) = 𝜂𝑖(0), there always exists a small value of 𝑐(such that 𝑥𝑖is small enough) and small magnitude of 𝜃 < 0to locally ensure the following: 𝛹𝑅(𝜃(𝜂𝑖−𝑥𝑖) + 𝑥𝑖)=𝜃(𝜂𝑖−𝑥𝑖) + 𝑥𝑖.(41) Under this condition and defining the tracking error 𝛿=𝜂𝑖−𝑥𝑖, that is, error in tracking the actual IC to the virtual IC, we obtain  𝛿=𝜃𝛿 under  𝛿=𝜂𝑖−𝑥𝑖and using (29). For 𝜃 < 0, it can be observed that the actual IC 𝜂𝑖(𝑡)will converge to the virtual IC 𝑥𝑖(𝑡)because 𝛿→0. Step III: Here, the combination of the results obtained in Steps I and II establishes the main condition 𝜀𝑖-KKT in Lemma 2, given by (27). By considering 𝑒𝑇(𝑡)𝑒(𝑡)≤𝜌∕(2𝑞𝑐𝜆min(𝛬𝐿) − 𝜗)𝑞from the results of Step I at the steady-state, using 𝑒𝑖(𝑡) = 𝑥𝑖(𝑡) − 𝑥 and 𝜂𝑖→𝑥𝑖for a local region (as in Step II), and selecting 𝜆=𝑥, it can be observed that the following relation will be attained at the steady-state. ‖‖𝜂𝑖(𝑡) − 𝜆‖‖≤√𝜌∕(2𝑞𝑐𝜆min(𝛬𝐿) − 𝜗)𝑞. (42) That is to say that there exists a scalar 𝜉such that 𝜂𝑖and 𝜂𝑗will converge to the region, given by ‖‖‖𝜂𝑖(𝑡) − 𝜂𝑗(𝑡)‖‖‖≤𝜉, 𝑖, 𝑗 = 1,…, 𝑁, 𝑖 ≠𝑗, (43) which is equivalent to the condition (27) in Lemma 2. Step IV: In Step IV, we validate the condition (19) in Lemma 2 through the proposed protocol under Assumption 1. Dividing (31) by 2𝑎𝑖and applying the summation, we have ∑𝑁 𝑖=1 𝑥𝑖(𝑡)∕2𝑎𝑖=𝑐 𝑁 ∑ 𝑖=1∑𝑁 𝑗=1𝛼𝑖𝑗 (𝑥𝑗(𝑡) − 𝑥𝑖(𝑡)) +𝑐∑𝑁 𝑖=1 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑠𝑖(𝑡) − 𝑠𝑗(𝑡)). (44) Note that the double summations on the right side will be zero, that is, to say 𝑁 ∑ 𝑖=1 𝑥𝑖(𝑡)∕2𝑎𝑖= 0.(45) Under initial condition 𝑥𝑖(0) = 𝜂𝑖(0), we have 𝑁 ∑ 𝑖=1 𝑥𝑖(𝑡)∕2𝑎𝑖= 𝑁 ∑ 𝑖=1 𝜂𝑖(0)∕2𝑎𝑖 = 𝑁 ∑ 𝑖=1 𝑃𝑖(0). (46) Under the convergence of 𝜂𝑖(𝑡)to 𝑥𝑖(𝑡)and under the assumption of initial supply–demand balance, it can be concluded that 𝑁 ∑ 𝑖=1 𝜂𝑖(𝑡)∕2𝑎𝑖= 𝑁 ∑ 𝑖=1 𝑃𝑖(𝑡)→ 𝑁 ∑ 𝑖=1 𝑃𝑖(0) = 𝑃𝐷+𝑃𝐿−𝑃𝑠,𝑚 −𝑃𝑤,𝑛,(47) which shows that the condition (19) in Lemma 2 is validated. Step V: In the final step, we validate the condition (20) of Lemma 2 for 𝜀𝑖-KKT condition. To validate (20), note that the function 𝛹𝑅(⋅)in (30) satisfies − 2𝑎𝑖𝐷𝑅𝑃 𝑖≤𝛹𝑅(⋅)≤2𝑎𝑖𝑈𝑅𝑃 𝑖.(48) Applying 𝜂𝑖=𝛹𝑅(𝜃(𝜂𝑖−𝑥𝑖)), we have − 2𝑎𝑖𝐷𝑅𝑃 𝑖≤𝜂𝑖(𝑡)≤2𝑎𝑖𝑈𝑅𝑃 𝑖.(49) Dividing by 2𝑎𝑖and using the fact that 𝜂𝑖(𝑡)=2𝑎𝑖𝑃𝑖(𝑡), we achieve (20) in the steady-state. As conditions (19)–(20) and (27) are validated, we achieve 𝜀𝑖-KKT condition of Lemma 2, which ends the proof of distributed optimization for ET EDP. In the next step, the elimination of Zeno behaviour through the proposed method will be demonstrated. Step VI: To show the elimination of Zeno behaviour, we construct the dynamics of triggering error 𝑠𝑖(𝑡). Taking derivative of 𝑠𝑖(𝑡) = 𝑥𝑖(𝑡)− 𝑥𝑖(𝑡𝑖 𝑘), applying 𝑥𝑖(𝑡)=2𝑐𝑎𝑖∑𝑁 𝑗=1 𝛼𝑖𝑗 (𝑥𝑗(𝑡𝑗 𝑘) − 𝑥𝑖(𝑡𝑖 𝑘)),𝑒𝑖(𝑡) = 𝑥𝑖(𝑡) − 𝑥 and 𝑥 =𝑁−1 ∑𝑁 𝑗=1 𝑥𝑗(𝑡), and considering bounded steady-state value of 𝑒𝑖(𝑡), we have ||𝑠𝑖(𝑡)||=|||||| 2𝑐𝑎𝑖 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑥𝑗(𝑡𝑗 𝑘) − 𝑥𝑖(𝑡𝑖 𝑘))|||||| = 2𝑐𝑎𝑖|||||| 𝑁 ∑ 𝑗=1 𝛼𝑖𝑗 (𝑒𝑗(𝑡𝑗 𝑘) − 𝑒𝑖(𝑡𝑖 𝑘))|||||| ≤𝑀, (50) where 𝑀is a bounded real positive scalar. For 𝑡∈[𝑡𝑖 𝑘, 𝑡𝑖 𝑘+1),||𝑠𝑖(𝑡)||≤ 𝑀ensures ||𝑠𝑖(𝑡)||≤𝑀(𝑡𝑖 𝑘+1 −𝑡𝑖 𝑘). Based on the triggering condition, we have ||𝑠𝑖(𝑡)||≥√𝜍𝑖, leading to √𝜍𝑖≤𝑀(𝑡𝑖 𝑘+1 −𝑡𝑖 𝑘). Hence, the interval between two successive events is strictly positive and is given by 𝑡𝑖 𝑘+1 −𝑡𝑖 𝑘≥√𝜍𝑖∕𝑀, leading to elimination of Zeno behaviour. It completes the proof of Theorem 1.■ Remark 6. It is important to note that the practical generation constraints (PGCs) in (12) and (13) must be considered throughout the power 𝑃∗ 𝑖allocation process, since these constraints have an influence on the solution of the optimization algorithm. PGCs are managed in some (distributed) studies by incorporating a logarithmic barrier function and a convex quadratic equation (Yang et al.,2022). Indeed, these penalty terms have the potential to address PGCs; nevertheless, constraint violations cannot be completely avoided during the convergence process, if the choice variables is not properly accounted. The flowchart of the proposed ET EDP approach, in order to provide the procedural steps, has been presented in Fig. 1. The flowchart consists of two distinct blocks, named as (i) Initialization and Requirement and (ii) Upgradation Protocols. The primary objective of Initialization and Requirement block is to obtain the pertinent system parameters. The other block Upgradation Protocols provides a recursive procedure to update the system protocol. This iterative process ensures improvement in power systems’ performance to achieve the desired requirements and objectives. Energy Reports 10 (2023) 4097–4111 4104 I. Ahmed et al. Fig. 1. Flowchart of distributed ET EDP approach. Remark 7. As mentioned earlier, the protocol (29) can be applied to resolve distributed EDP under RRLs compared to the existing distributed methods due to 𝜂𝑖=𝛹𝑅(𝜃(𝜂𝑖−𝑥𝑖) + 𝑥𝑖)and saturation function in (30). The convergence analysis is quite challenging compared to Bai et al. (2019), Gao et al. (2021), Xu et al. (2021a), Li et al. (2015), Wan et al. (2021), Wang et al. (2021b) and Liu et al. (2018) in this scenario, which is addressed in the proof of Theorem 1 through a five-step approach. Various constraints are ensured through virtual IC consensus, convergence of actual ICs to the virtual ones, practical consensus between actual ICs, tracking of supply–demand balance, enforcement of RRLs through the saturation function. Remark 8. It should also be noted that the mentioned problem of distributed EDP under RRL constraint has been solved with ET mechanism in comparison to He et al. (2018), Bai et al. (2019), Xu et al. (2021a), Wang et al. (2021b) and Liu et al. (2018). The proposed approach ensures elimination of Zeno behaviour through a computationally simple triggering mechanism with respect to Wan et al. (2021). Remark 9. In contrast to the existing studies outlined in Refs. Wu et al. (2022), Gafar et al. (2022) and Ahmed et al. (2022e), the current research work employs an advanced distributed protocol with the objective of achieving optimal IC consensus (please see Table 1). The proposed distributed communication and consensus protocol enables the system to continue functioning in the event of node failures, ensuring the overall integrity of a system. Moreover, in contrast to the findings in Xu et al. (2021a), Wang et al. (2021b) and Liu et al. (2018), the proposed model incorporates GPSs, which effectively deals with the carbon emissions, produced by thermal plants, by facilitating the sharing of energy grid loads. This integration of GPSs into the proposed model advances the mentioned approach for sustainable energy practices. Additionally, the devised protocol also incorporates an ET condition for communication, to achieve fewer use of communication bandwidth in comparison to Bai et al. (2019). By leveraging these features, the resultant model achieves efficiency and sustainability while maintaining the communication reliability and resource allocation (please see Table 1). 5. RETScreen software and meteorology information (solar and wind data) RETScreen is based on Microsoft Excel, also known as RETScreen Clean Energy Project Analysing Software. It is a tool that utilizes analytical techniques to assess GPSs (Lee et al.,2012). The software was developed in 1996 at Canada Canmet Energy Research Center and was ultimately selected because of its ability to perform multidimensional assessments of GPSs finance, risk evaluation, and greenhouse gas emissions. The mentioned software has an extensive and detailed climatology database. The RETScreen Meteorological Library offers a wide range of solar and wind data, which can be employed for attaining the forecasting models. It offers access to climate-related Energy Reports 10 (2023) 4097–4111 4105 I. Ahmed et al. Table 1 Feature comparison of proposed work. Works Comparison features Year RRLs Nonlinear dynamics Distributed protocols ET Condition GPSs Integration Wu et al. (2022), Gafar et al. (2022) and Ahmed et al. (2022e) 2022 ✓ ✓ × × × Bai et al. (2019), Liu and Song (2022) and Li et al. (2015a) 2019 2021 ×✓ ✓ × × Güvenç and Kaymaz (2019) 2019 2018 ×✓ ✓ × × Rauf et al. (2021) 2021 ×✓ ✓ ✓ × Proposed approach ✓ ✓ ✓ ✓ ✓ information, obtained from approximately 6700 local observation stations (Milosavljević et al.,2022). It also allows users to access precise weather-related data for analysis and evaluation purposes. The RETScreen Meteorological Database can provide two main sources of information. First, it provides the information gathered from various local stations around the globe. These meteorological stations collect data of solar radiation, wind velocity, and humidity. The acquired data offers a valuable insight on the weather conditions and facilitates accurate research and assessing on green energy systems. Second, the database also contains the data from satellites under the jurisdiction of NASA (Ahmed et al.,2023b,2022a). The RETScreen offers a range of tools to users for evaluating the potential energy savings, greenhouse gas emissions, and financial feasibility of sustainable energy initiatives. This software facilitates the decision-making process for individuals and organizations (Carvalheira et al.,2023). It maintains the data integrity which strengthens its reputation, hence allowing stakeholders to make educated decisions, based on trustworthy information (Luo et al.,2023). Some of the detailed benefits of using RETScreen software are as follows: 1. Energy Project Feasibility Analysis: The RETScreen software enables the examination of initiatives related to renewable energy. The system allows users to input project details and obtain analyses on energy generation, consumption, and savings. 2. Accurate Weather Data: The software provides the real-time climate data, obtained from meteorological stations, satellites, and climate models, which is crucial for attaining precise energy results. 3. Financial Assessment: The assessment of the financial feasibility of projects can be attained through several factors such as expenses and revenues. The present work has employed the meteorological data from wind and solar sources to determine the ED using GPSs. Table 2 depicts the annual climatic data statistics of solar and wind, obtained from RETScreen. The site location of GPSs sources is 40◦S 146◦E in Bass Strait. Fig. 2 shows the yearly wind velocity and sun irradiation profile. The conventional thermal power plants can generate electricity independent of atmospheric conditions. In contrast, GPSs heavily rely on the climate, for which RETScreen can provide the useful information. Consequently, the authors have employed 𝐹𝛽(𝑀)technique to incorporate the uncertainty pertaining to GPSs. The beta probability density function (PDF) model has been applied with range of 0 to 1. The solar irradiance is treated as a stochastic process through the adoption of the beta distribution by taking 𝐹𝛽(𝑀)as a random variable. To obtain the power generated by solar panels, we can apply 𝑃𝑠,𝑚 =𝑃𝑠0×𝐹𝛽(𝑀), which can be integrated into the solar cost function to compute the overall solar cost. 6. Results and simulations The efficacy of the proposed ET EDP methodology has been assessed through the application of two distinct case studies, one involving the presence of GPSs and the other without GPSs. The case study without Fig. 2. Annual climatic data plot. GPSs is employed to assess the conventional EDP approach for fuelbased power generators. While the case study with GPSs has been applied to consider integration of renewable sources, for attaining the environmental benefits like lower emissions (Pedersen et al.,2022), enhanced energy security (Kasradze et al.,2023), employment creation (Asmelash et al.,2020), and policy insights. These case studies are conducted on a network, consisting of six generators, arranged in a ring configuration. The diagram illustrating the experimental design of the proposed network is shown in Fig. 3. 6.1. Case study-I (ET ED problem without GPSs) This case study demonstrates the feasibility of implementing the proposed distributed ET EDP approach in the absence of GPSs. The tabular representation, provided in Table 3, shows the parameters for the fuel cost functions, along with the RRL constraint limits (AlRoomi,2016). To meet the smart grid load demand presented in (12), the following initial conditions on generation are assumed for all the committed units. 𝑃1(0) = 300MW, 𝑃2(0) = 200MW, 𝑃3(0) = 300MW, 𝑃4(0) = 150MW, 𝑃5(0) = 200MW, 𝑃6(0) = 120MW.