1.1. Problem Definition Small errors in lifetime prediction can lead to significant financial uncertainty, as a 2% underestimation of degradation results in >10% distortion in the levelized cost of storage [ 5]. Hence, credible degradation models in real-time are necessary for maintenance planning but also for operational optimization and grid-service bidding. Various modeling techniques have been developed to describe degradation, but they still suffer from significant constraints. Empirical models rely on curve-fitted relations of capacity fading with operational stress variables, such as depth-of-discharge ( DoD), C-rate, and temperature [ 6]. These models are computationally cheap but are often valid only in the particular laboratory settings for which they were tuned. At the other end of the spectrum, physics-based models, most notably the Doyle–Fuller–Newman (DFN) framework [ 7], offer deep insight into lithium transport, Solid Electrolyte Interphase (SEI) growth, and reaction kinetics, but require parameters that are difficult to measure in commercial cells and impose computational burdens that are incompatible with real-time control. 1.2. Research Motivation and Gap Battery Energy Storage Systems play a critical role in renewable energy integration, peak shaving, and electricity market participation. However, their economic viability depends not only on operational revenues but also on battery degradation, which directly affects lifetime and replacement costs. Therefore, accurate degradation-aware operational planning is essential for maximizing both profitability and asset longevity. Although significant progress has been made in battery degradation modeling and BESS optimization, many existing approaches rely on fixed model parameters and simplified degradation assumptions that do not reflect the evolving condition of batteries during operation. Similarly, most scheduling frameworks optimize energy dispatch without continuously incorporating updated battery health information into decision-making. To address this gap, this study proposes an SLDT framework that combines an ARDM [ 9] with an optimization and control layer. The proposed framework continuously updates battery degradation parameters using operational data and incorporates the updated degradation information into optimal scheduling decisions under dynamic electricity tariffs. The main contributions of this work are as follows: • Development of a self-learning digital twin framework for lithium-ion BESSs based on adaptive degradation estimation. • Integration of real-time degradation prediction with optimization-based operational scheduling. • Comparative evaluation of fixed and dynamic tariff schemes using degradation-aware economic analysis. • Experimental validation using commercial lithium-ion battery modules under various operating conditions. 1.3. Objectives and Contributions A third aim is to integrate the DT with an optimization and control layer that provides degradation-aware charge/discharge trajectories based on day-ahead pricing signals. This is in line with recent scheduling approaches that include degradation in the dispatch decisions [ 23, 24]. To enable real-time deployment, the framework considers numerous meta-heuristic optimization algorithms with particular interest, such as the Musical Chairs Algorithm (MCA) [ 25], whose fast convergence has been verified in several applications [ 25, 26, 27, 28]. To examine the performance in the context of multi-variable scheduling constraints of the BESS, comparative evaluations with swarm-reduction PSO and GWO versions [ 29, 30] are provided. 1.4. Organization of the Paper 2.1. Battery Degradation Modeling Approaches Empirical models establish mathematical relationships between capacity fade and operating variables such as temperature, DoD, SoC, and C-rate [ 9, 37]. These models are computationally efficient and suitable for system-level applications; however, their predictive capability is often limited to operating conditions represented in the calibration dataset. Semi-empirical approaches attempt to balance accuracy and computational efficiency by incorporating degradation mechanisms into simplified mathematical structures. Several studies have demonstrated that semi-empirical models can achieve reliable degradation prediction while maintaining computational tractability for online applications [ 9, 37 Despite significant progress, many degradation models rely on fixed parameters identified offline and do not continuously adapt to changing battery conditions throughout the battery lifecycle. 2.2. Physics-Based and Reduced-Order Models While the semi-empirical structure theoretically allows adaptation to other chemistries (e.g., NMC, LCO) by updating ARDM parameters, the framework is currently validated only on LFP modules. Validation across multiple chemistries is reserved for future work. 2.3. Data-Driven and Machine-Learning Approaches Advances in machine learning (ML) and adaptive estimation techniques have enabled the development of data-driven battery models capable of capturing nonlinear degradation behavior [ 17]. Methods such as artificial neural networks [ 15, 21], support vector machines [ 43], Gaussian process regression [ 17], recursive least squares [ 44], and deep learning architectures [ 45] have been applied to battery health prediction. Methods used are Gaussian-process regression (GPR) [ 17], random forests [ 19], gradient boosting [ 20], and recurrent neural networks (RNN) [ 21]. Early-cycle impedance or voltage characteristics can be used with deep-learning frameworks to forecast residual usable life (RUL) [ 22]. For instance, Severson et al. [ 17] were able to estimate longevity with 90% accuracy using just 100 cycles, while Attia et al. [ 46] used active learning to minimize the number of tests by 80%. Despite these advances, solely data-driven models have limited extrapolation potential, incorrect outputs under specific operating situations, and lack physical transparency. Differential-equation-constrained learning frameworks are embedded by hybrid or physics-informed neural networks (PINNs) [ 15]. Han et al. [ 15] used PINNs to predict the SoH in real time, achieving 0.65% error, but required GPU acceleration. The challenge remains to merge such learning capability with adaptive physical models executable on industrial controllers. Adaptive models are particularly attractive because they can continuously update model parameters as new operational data become available [ 9, 47]. Such capability is important for batteries operating under highly dynamic conditions where degradation behavior evolves over time. 2.4. Digital Twin Technologies for Battery Systems Digital twin technology has emerged as a key enabler of intelligent asset management in Industry 4.0 environments, enabling real-time monitoring, prediction, optimization, and decision-making through continuous interaction between physical and virtual systems [ 48]. A digital twin is generally defined as a virtual representation of a physical system that continuously exchanges information with its physical counterpart to support monitoring, prediction, optimization, and decision-making [ 49 However, many battery digital twin studies primarily focus on condition monitoring and health prediction. Relatively few investigations integrate degradation-aware optimization and operational scheduling within the digital twin framework. Furthermore, practical implementations capable of continuously updating degradation model parameters using operational data remain limited. 2.5. Optimization Algorithms for Parameter Estimation Optimization plays a critical role in maximizing the economic value of battery energy storage systems. Existing scheduling approaches have been developed for energy arbitrage, peak demand reduction, frequency regulation, renewable energy integration, and microgrid management. Although considerable progress has been achieved, many optimization studies employ simplified degradation assumptions or neglect degradation costs entirely. Such simplifications may produce economically attractive short-term schedules but can underestimate long-term battery replacement costs and lifetime impacts. 2.6. Research Gap and Positioning of the Present Work The literature review demonstrates significant progress in battery degradation modeling, digital twin architectures, and optimization-based BESS scheduling. However, several critical challenges remain unresolved in the existing literature, which typically addresses these domains in an isolated manner. To clearly articulate the positioning of this study and highlight the gaps it bridges, a structural comparison with foundational approaches is formalized in . The deficiencies of the current state of the art can be categorized into three fundamental technical gaps: Static Parameterization in Dynamic Environments: Many conventional degradation models rely on fixed parameters identified offline through time-consuming laboratory pre-cycling. While computationally efficient, these frameworks fail to continuously adapt to the evolving structural and chemical conditions of the battery during real-world operations. Consequently, their predictive fidelity severely deteriorates when cells are exposed to highly dynamic operational variations in temperature, DoD, and C-rate over long-term field deployment. Passive Monitoring vs. Active Prescriptive Control: Existing battery digital twins predominantly function as passive diagnostic platforms. They excel at condition monitoring, SoC estimation, fault diagnosis, and RUL forecasting. However, while these capabilities vastly improve asset visibility, they lack prescriptive agency; most reported digital twin frameworks do not feed these real-time health insights back into an active, optimization-based control loop to alter the charge/discharge trajectories of the physical asset. Misalignment of Economic Dispatch and Physical Degradation Costs: Optimization frameworks frequently evaluate energy arbitrage and grid scheduling by assuming simplified, linear degradation penalties or arbitrary, constant degradation costs. Such oversimplifications fail to capture the highly nonlinear aging dynamics governed by physical stress variables. As a result, the resulting economic evaluations introduce severe financial uncertainty, failing to accurately represent the critical trade-off between short-term revenue generation and long-term battery health preservation. The authors’ baseline framework, the ARDM [ 9], was originally conceived to mitigate some of these challenges by enabling online degradation estimation through continuous parameter updating via operational data. However, as mapped in , ARDM was strictly restricted to a reactive degradation prediction role. It operated autonomously without an overarching digital twin infrastructure, was completely decoupled from real-time operational optimization layers, and lacked any economic dispatch or dynamic market bidding functionality. To overcome these fragmented limitations, this study introduces a comprehensive, closed-loop SLDT framework that expands the predictive capability of the ARDM into a prescriptive energy management platform. By integrating the self-learning ARDM engine, a real-time synchronized virtual twin layer, and an active OCL driven by the high-convergence MCA, the proposed architecture accomplishes what isolated existing studies cannot. As contrasted in , ref. [ 25] demonstrates strong credentials in health-aware scheduling optimization and economic dispatch under dynamic conditions, yet operates with static degradation parameters and entirely lacks a synchronized virtual asset representation. On the other hand, traditional digital twin frameworks, represented by Ref. [ 14], focus heavily on establishing virtual-to-physical synchronization for health diagnostics but lack an integrated operational optimization engine or market-bidding economic layer. By successfully uniting adaptive degradation estimation, continuous digital twin synchronization, metaheuristic scheduling, and techno-economic validation within a single computing environment, the proposed SLDT resolves these long-standing contradictions. This framework transitions the BESS from a passively monitored asset to an intelligently optimized, health-aware participant in dynamic electricity market structures. 3. Methodology The technique proposed in this paper employs the ARDM [ 9] to estimate the degradation cost based on real operation with the best accuracy, and then uses the SLDT to estimate the optimal contribution of the BESS for the largest revenue. The approach is made up of three separate layers: the ARDM layer, the SLDT layer, and the OCL. The current SLDT implementation operates as a software-in-the-loop framework where experimental BESS data is streamed to the ARDM. A true closed-loop HIL deployment with real-time actuation feedback is recognized as a future research step. The synchronization frequency is set to update parameters every 200 samples, and the data flow follows: Physical BESS → DAQ → ARDM Parameter Estimator → OCL Day-Ahead Scheduler, as shown in . The control system SLDT is designed with two separate objective functions. The first one is the one used with ARDM to find the ideal degradation model parameters for the smallest RMSE between the measured and the calculated degradation of the BESS, to be used subsequently to compute the real-time degradation cost of the BESS. The second objective function is used to determine the optimal C-rate that the BESS can exchange with the grid for the highest revenue according to the day-ahead tariff. The block diagram showing the layers of the proposed system is shown in . In the following subsections, detailed descriptions of these two different objective functions are provided. 3.1. Adaptive Real-Time Degradation Modeling (ARDM) A semi-empirical degradation model ARDM [ 9] is used to determine the optimal degradation model parameters for minimum RMSE between the measured and calculated degradation as indicated in Equation (1). This part extracts the degradation model parameters so that the predicted degradation level ( D E G c d ) closely matches the measured one ( D E G m d ). This objective function determines the optimal degradation model parameters for minimum RMSE between the measured capacity from the actual operation of the system and the predicted (calculated) values from the introduced model, as shown in Equation (1) [ 9R M S E ( α , β , ψ , ζ ) = 1 / N ୍ଠ ∑ d = 1 N D E G m d − D E G c d 2 (1) The degradation per sampling time, as shown in , can be obtained as shown in Equation (2) [ 9D E G r a m p = 0.5 N c . 1 − D int − D e n d D e n d ζ (2) where Dint, Dend are the initial and end DoD of the ramp, and Nc is the maximum number of lifetime cycles that can be obtained from Equation (3) [ 37N c D , P B i , T = L T C ( D 100 % ) θ D , P B i , T = L T C ( D 100 % ) . D D 100 % − 1 / α C r a t e t − 1 / β ୍ଠ e − ψ 1 T R − 1 T i (3) where LTC lifetime cycles at DoD = D100%, which can be obtained from the Wöhler curve at 100% DoD [ 9 Substituting the value of Nc obtained from Equation (3) into the degradation per ramp equation shown in Equation (2), the degradation per ramp can be obtained as shown in Equation (4) [ 23, 24]. The total degradation per day is the summation of the degradations per sample during the day, which can be obtained from Equation (5) [ 23, 24D E G r a m p = Δ t L T C ( D 100 % ) D D 100 % 1 / α 2 · C r a t e t 1 / β ୍ଠ e ψ 1 T R − 1 T i (4) D E G d a y = 1 6 ∑ i = 1 N s D E G r a m p i (5) where Δ t is the duration time of the ramp in hours, and Ns is the total number of ramps (samples) per day. As an example, if the sampling time is one hour, then Ns is 24 samples; meanwhile, if the sampling time is 10 min, then the number of samples is 144. The flowchart of the logic used with ARDM is shown in . In this flowchart, the optimization algorithm sends the initial values of degradation model parameters and the value of LTC ( DoD = 100%), which can be obtained from the Wöhler curve that comes in the datasheet of the batteries. The degradation per ramp (sample) can be obtained from Equation (4) for each sample. After each sample, the logic moves to the next one and accumulates the degradations to the end of the samples per day to determine the daily calculated degradation D E G c d . The RMSE is determined as shown in based on Equation (1). This value goes back to the optimization algorithm as a fitness value for the input parameter values sent to the objective function. The flowchart showing the logic of MCA and how it can be used with ARDM or any other optimization algorithm is shown in [ 25 3.2. Self-Learning Digital Twin Architecture The proposed SLDT framework establishes a continuous interaction between the physical BESS and its virtual counterpart to support degradation-aware operational decision-making. The architecture consists of five interconnected layers: (i) physical battery system, (ii) data acquisition layer, (iii) ARDM, (iv) OCL, and (v) operational feedback layer. The physical layer consists of the battery modules operating under different C-rate, DoD, and temperature conditions. During operation, battery voltage, current, temperature, SoC, and capacity-fade information are collected through the data acquisition system. These measurements are continuously transferred to the virtual environment, where they are processed by the ARDM. The ARDM acts as the core predictive engine of the digital twin. Every 200 samples, the model updates its degradation parameters using the latest operational measurements. This adaptive updating mechanism enables the digital twin to continuously synchronize its degradation state with the actual condition of the physical battery and to accurately estimate future degradation behavior under changing operating conditions. The updated degradation information is then transmitted to the OCL, where battery dispatch decisions are determined based on electricity tariffs, battery operating constraints, and degradation costs. The optimization process generates the optimal charging and discharging schedule that maximizes economic performance while limiting degradation-related losses. Finally, the optimization results are fed back to the digital twin environment to update the predicted battery state and operational strategy for the next scheduling interval. Through this closed information loop, the SLDT continuously adapts its internal model parameters and operational decisions according to the latest battery condition and market signals. It should be noted that the current implementation represents a software-in-the-loop (SIL) digital twin framework. The synchronization between the physical and virtual systems is achieved using experimentally acquired battery data rather than a fully deployed hardware-in-the-loop or field-scale implementation. Nevertheless, the proposed architecture establishes the foundation for future real-time deployment in practical grid-connected BESS applications. 3.3. Optimization and Control Layer (OCL) The second objective function is to maximize the revenue from the BESS, as shown in Equation (6). The choice of revenue to be maximized will also ensure the prolongation of the battery life and maximize the operation of the BESS. The second term in the multi-objective function is to minimize the change difference between the SoC of the BESS at the start and end of the day. This objective function will be performed once a day to choose the plan of the contribution from the BESS throughout the day (the value and time of the C-rate). The sampling time determines the number of variables used in optimization, where the number of variables is 24 and 144 in the case of one-hour and 10 min sampling, respectively. The sampling time can be chosen based on the available data from the system, where the system should provide the SLDT with the tariff. The variation in the tariff can simulate the requirements from the BESS, where, in the case of the power system needing energy from the BESS, the tariff should be high; conversely, the tariff will be low when the power system needs the BESS to extract the surplus power. There should be constraints when applying this objective function, as shown in Equations (7)–(9). The logic used in the OCL is shown in . J C r a t e ( t ) = max ( w 1 ⋅ R n e t / d a y − w 2 Δ S o C ) (6) where, w1w2 are weight factors subject to w1 + w2 = 1. The objective function Equation (6) is subject to safety and grid constraints as shown in the following: S o C min ≤ S o C ( t ) ≤ S o C max (7) C r a t e ( t ) ≤ C r a t e max (8) T ( t ) ≤ T max (9) 3.4. The Economic Study 3.4.1. Fixed Tariff Model for Private BESS Under the Saudi Electricity Company (SEC) Fixed or Static Tariff Model (FTM) represents one of the most practical and reliable mechanisms for enabling BESSs’ participation in grid operations managed by the Saudi Electricity Company (SEC) [ 31]. This model provides a predictable and regulated payment system that allows for early-stage BESS investment while preserving operational control by the national utility. Under the FTM, the remuneration that a private BESS’s owner receives is based on predefined prices for capacity availability, energy services, and ancillary support. Unlike dynamic or market-based systems, where tariffs fluctuate according to instantaneous supply-demand conditions, the fixed tariff approach relies on a regulated pricing structure approved by the Water and Electricity Regulatory Authority (WERA), formerly ECRA [ 32]. This provides stability and transparency for the utility and the investors. The BESS is operated according to schedule or control instructions from the SEC. Remuneration is established at fixed rates, as opposed to market bids, in order to deliver services such as peak shaving, load balancing, and frequency regulation. To address methodological asymmetry, it is clarified that FTM relies on a contracted fixed degradation rate assuming a baseline dispatch, whereas DTM reflects the true dynamic degradation. Both models are subjected to the exact same operational constraints. The features of the fixed tariff structure make it particularly ideal for early or transitional periods of BESS deployment. Its simplicity and predictability foster involvement from private investors who would otherwise be reluctant to enter a young energy market. The approach usually requires monthly or quarterly payments that are made up of three main components: a capacity payment, a performance-based energy payment, and an energy arbitrage payment. The capacity payment compensates the BESS for being available to provide grid services, the performance payment compensates for the actual energy delivered or absorbed during grid events, and the arbitrage payment compensates for scheduled energy transfers between off-peak and on-peak periods at fixed price differentials. These are contractually specified between the BESS operator and the SEC to ensure financial certainty and to simplify the billing process. In terms of mathematics, the total monthly revenue of the BESS ( Rtotal) can be calculated as the sum of the capacity payment ( Rcap), performance payment ( Rperf), and arbitrage revenue ( Rarb), less the total operational expenses ( Ctotal), as illustrated in Equation (10). R t o t a l = R c a p + R p e r f + R a r b − C t o t a l (10) where • Rcap: Capacity payment • Rperf: Performance-based energy payment • Rarb: Energy arbitrage revenue • Ctotal: Total costs due to degradation, operation, and maintenance The capacity payment can be obtained from Equation (11) R c a p = P r a t e d · A v · C F R , c a p (11) where: • Prated: Rated BESS power (kW) • Av: Availability factor. “It indicates how often a plant or system is technically ready for operation compared to the total time.” • CFR,cap: Fixed capacity tariff (SAR/kW/month) The performance-based energy revenue can be obtained from Equation (12) R p e r f = E F R , d e l · C F R , p e r f (12) where: • EFR,del: Actual MWh delivered for frequency support • CFR,perf: Payment rate for delivered energy for frequency support (SAR/MWh) The arbitrage revenue can be obtained from Equation (13). R a r b = N c y c l e s · E u s a b l e · D · η · Δ λ (13) where: • Ncycles: Average cycles per day × days/month • Eusable: Usable energy capacity (kWh) • η: Round-trip efficiency • Δ λ: Fixed arbitrage price spread (SAR/kWh) between off-peak and peak hours • D: The allowable DoD The total costs due to degradation, operation, and maintenance can be determined from Equation (14). C D & O & M = C O & M + C deg (14) where: • CO&M: Monthly fixed operations and maintenance cost • Echarge: Energy consumed during charging (kWh) • Poff-peak: Off-peak energy price (SAR/kWh) • Cdeg: Degradation-related cost due to cycling Each of these components can be calculated based on system capacity, availability, efficiency, and fixed tariff rates. For instance, the capacity payment depends on the rated power of the system, its availability factor, and a fixed monthly tariff. The performance-based component depends on the energy actually delivered during frequency regulation events multiplied by a fixed performance rate. Similarly, the arbitrage revenue depends on the usable energy capacity, the number of operating cycles per month, the system’s round-trip efficiency, and the fixed spread between off-peak and peak tariffs. The FTM has several advantages. And, most crucially, it offers investors revenue stability, which enables them to more easily arrange funding and evaluate financial returns with confidence. The concept significantly streamlines administrative oversight by allowing the SEC to directly regulate dispatch schedules and guarantee that BESS functioning is in accordance with system requirements. Additionally, the model mitigates the risk of uncoordinated dispatching that could compromise grid stability by preventing exposure to real-time price variations. This approach is especially pertinent for pilot projects or government-sponsored demonstration programs that are designed to acquire operational experience with BESS before the introduction of competitive markets. The approach is not without its limitations. In a system with volatile renewable generation, fixed rates may not completely capture the economic benefits of variable energy storage. They are designed to prohibit BESS operators from exploiting large price spreads or responding appropriately to quickly changing system conditions. Therefore, the total economic efficiency generated by the model may be less than DTM. It can also be an obstacle to innovation, as operators are incentivized based on set rates, not strategic dispatch or performance excellence. Moreover, the fixed tariff structures may not reflect the degradation costs, which could affect the system stability over time [ 33]. Therefore, a fixed tariff regime is a transitional paradigm that allows the regulated involvement of BESS while the legal and technological pillars for DTM are being developed. Furthermore, this methodology is supported by analogous cases from other countries. At the outset, the Enhanced Frequency Response (EFR) program in the United Kingdom (2016–2018) implemented fixed-rate contracts to encourage the deployment of batteries. Conversely, it eventually converted to an auction-based DTM [ 34]. In the same line, the storage initiatives of the Dubai Electricity and Water Authority (DEWA) and the early California ISO (CAISO) storage pilots (2014–2016) [ 34]. Before the integration of BESS into energy markets, fixed compensation schemes were developed to obtain operating expertise [ 51]. As these worldwide examples suggest, FTMs are a realistic intermediate step in the pursuit of a more competitive, value-based participation paradigm. The FTM paves the way for early deployment of BESS under the umbrella of Saudi Vision 2030, which stresses the diversification of renewable energy and the upgrading of the grid [ 31]. It enables the regulator to collect operational data for future market changes, improves grid stability amid increasing solar PV adoption, and facilitates coordination between the SEC and private investors [ 32]. By combining technical reliability with financial predictability, the model acts as a bridge between today’s regulated system and tomorrow’s dynamic, market-driven energy future. The Kingdom of Saudi Arabia is working to expand its energy-storage capacity through BESS to reach 48 GWh by 2030 [ 31]. This represents the largest project of its kind in the GCC region, ensuring grid stability and maximizing the utilization of renewable energy sources [ 31 3.4.2. Dynamic Tariff Model for Private BESS Under ARDM The revenue calculation with the DTM will have the same calculation as the FTM for the capacity payment Rcap and the performance-based energy revenue, Rperf, for a fair comparison. Meanwhile, the arbitrage revenue, Rarb, and degradation, operation, and maintenance costs, CD&O&M, are different, as will be discussed in the following: The arbitrage revenue, Rarb, can be determined by subtracting the charging cost from the discharge income as shown in Equation (15). R a r b = I a r b − C a r b (15) where Iarb: Monthly discharging income which can be obtained from Equation (16) Carb: Monthly charging cost of the battery, which can be obtained from Equation (17) I a r b = 30 6 ୍ଠ ∑ t = 1 144 λ t · η d · C r a t e t · E R where C r a t e t 0 (17) where ηd is the discharging efficiency of the BESS. ηc is the charging efficiency of the BESS. The total costs due to degradation, operation, and maintenance can be determined from Equation (14). The operation and maintenance cost is the same as FTM for a fair comparison. Meanwhile, the degradation cost should be determined based on the real operation from the ARDM, in which the degradation per day can be obtained from Equation (5) and use this value to calculate the degradation cost per month as shown in Equation (18). C t o t a l = 30 · D E G d a y · ( C b − C E o L ) / ( 1 − Q E o L ) (18) where Cb is the total cost of the batteries, CEoL is the end-of-life price of the retired battery, and QEoL is the relative capacity or SoH of the battery at which it is retired. 3.5. Optimization Algorithm Two objective functions in the system should be optimized at the beginning of the day. The first objective function is the one used in ARDM to determine the optimal degradation model parameters ( α, β, ψ, ζ) based on minimizing the RMSE between the measured and calculated degradation during the day. The second need for the optimization problem is selecting the charging/discharging power to/from the BESS during the day-ahead for the highest revenue. As has been discussed above, these two optimization problems are very complex, and they need an accurate and fast optimization algorithm. Based on the free-lunch theory [ 52], there is a trade-off between the accuracy and the convergence speed. One of the most important techniques that avoids this limitation in the optimization algorithms is to use a high number of search agents at the beginning of optimization to enhance its exploration and gradually reduce this number with the progress of the optimization iterations to enhance its exploitation and reduce the convergence time. This novel idea is first introduced in the MCA as one of the most effective optimization algorithms [ 50]. This algorithm has been examined in many applications that need high accuracy and fast convergence time, such as MPPT of PV systems [ 26], optimal operation of hybrid renewable energy systems [ 27], electric vehicle (EV) optimal dispatch strategy [ 25], PV-cells parameters estimation [ 28], etc. The operation of the MCA in different applications showed its superiority in reducing the convergence time to 10% of the time required for conventional optimization algorithms such as PSO and GWO [ 50]. So, it is not a good idea to compare it here with these conventional optimization algorithms. The idea here is to modify these conventional optimization algorithms to work with the same idea, starting with a high number of search agents and gradually reducing this number with the progress of optimization iterations for a fair comparison. The application of this modification to the PSO and GWO is called SRPSO [ 29] and SRGWO [ 30]. Similarly, the OCL needs to manage the contribution power daily with 144 variables, which surely adds a huge challenge to the optimization algorithms (MCA, SRPSO, and SRGWO). 3.5.1. The Mechanism of the Musical Chairs Algorithm (MCA) The MCA is a relatively novel, population-based metaheuristic technique designed to achieve a superior balance between exploration and exploitation. This equilibrium is established by a dynamic, iterative process modeled on the familiar game of musical chairs. At its foundation, MCA starts with a large initial population of potential solutions (sometimes called “players” or “search agents”). The starting size guarantees that the whole search space is explored thoroughly, which provides for exploration and decreases the chance of the algorithm becoming locked in a local optimum. The primary characteristic of MCA is the gradual, competitive reduction in swarm size through cycles. The algorithm first initializes the positions of the players, who represent different solutions, and subsequently evaluates their “fitness” with respect to the objective function. In each round (iteration), the number of available “chairs” is one less than the number of “players”. All players attempt to occupy the nearest chair, which is often a proxy for the current best-known solution or a leading solution within a local neighborhood. The worst-performing solution (the “loser” who fails to secure a chair) is eliminated from the population. Both the loser and the chair are removed before the next round. The remaining players update their positions (refining their solutions) based on their newly assigned “chair” and their distance to it. Random search components, including a Lévy flight distribution, are incorporated into the movement to prevent premature convergence by balancing the local search with a global one. The population is progressively reduced as the iterations go by, through the repetitive removal of weaker solutions. Such a dynamic size reduction assures that the computational power of the remaining high-fitness agents is only spent on the most promising regions of the solution space, thereby successfully shifting the focus of the algorithm. The dynamic modulation of the elimination rate enables a transition from a broad search (exploration) to a focused refinement (exploitation) that is unique to fixed-population algorithms. This mechanism ensures that MCA maintains strong exploratory capability early in the optimization process when the global optimum is unknown, yet rapidly accelerates to high-precision exploitation in the final stages. The flowchart showing this logic is shown in [ 25 3.5.2. Swarm Reduction Strategies in PSO and GWO While the MCA formalizes the concept of swarm size reduction, this strategy has also been independently adopted and integrated into improved versions of established metaheuristic techniques, notably the PSO and GWO. A rigorous comparison necessitates positioning MCA against these similar evolutionary improvements [ 29, 30 (A). Swarm Reduction in PSO (SRPSO) Standard PSO, famous for its simplicity and speed, relies on the constant interaction of all particles, each influencing the swarm based on its individual best position and the global best position. When this fixed population struggles to escape a local optimum, it is often a sign of insufficient exploration. Conversely, once the global region is found, the large population slows convergence. SRPSO strategies address this by either periodically restarting a portion of the swarm in a new area (re-initialization) or, more relevant to MCA, gradually reducing the overall number of particles [ 29]. These reduction methods often aim to eliminate the particles with the lowest fitness or those spatially clustered far from the global best. Unlike MCA, which fundamentally designs its iteration logic around population reduction, SRPSO typically adds the pruning mechanism as an external module to the core PSO equations. Comparative studies must therefore ascertain if MCA’s inherent, game-theoretic architecture yields a more optimal and less-parameter-dependent control over the exploration-exploitation balance than modifications applied to the traditional PSO framework. Comparing MCA against these swarm size reductions in GWO and PSO will be crucial for establishing its unique contribution and demonstrating its potential superiority for the computationally intensive task of BESS optimization. (B). Swarm Reduction in GWO (SRGWO) The standard GWO models the social hierarchy to drive search agents (alpha, beta, delta, and omega wolves) toward an optimal solution. However, maintaining a large, fixed omega population throughout the search can lead to excessive, non-productive exploration in later iterations. To counter this, SRGWO variants have been introduced [ 30]. These improved models employ a mechanism to prune the less-fit “omega” agents analogous to the “loser” elimination in MCA as the search progresses. The objective is to shift the balance from exploration to exploitation by dedicating computational resources only to the most promising search trajectories. The control mechanism is the focal point of the distinction; in contrast to MCA, which implements a competition-and-elimination dynamic, SRGWO often implements a fitness-based threshold or a fixed-rate reduction function that is associated with the maximum iteration count. A comparison would evaluate whether MCA’s competitive, round-by-round elimination offers a more adaptive and superior reduction in search redundancy than the more formulaic technique of SRGWO. 4. Experimental Work The experimental work of this study has fully demonstrated the accuracy, flexibility, and real-time learning abilities of the proposed ARDM. The main goal of this experimental framework, combined with basic manufacturer datasheet information, was to demonstrate that ARDM can accurately predict LIBs’ capacity degradation from simple operating data such as current, voltage, temperature, and SoC. The technique sought to achieve a highly scalable and chemistry-adaptable degradation model that can be quickly implemented in large-scale BESS by removing dependence on laboratory pre-cycling testing or chemistry-specific calibration procedures. To ensure that the validation approach adequately reflected both controlled and actual field scenarios, two commercial 48 V LIB modules, Bat-1 and Bat-2, were selected as representative samples. Two commercial 48 V LIB modules, Bat-1 and Bat-2, were selected as representative samples to ensure that the validation approach appropriately represented both controlled and actual field situations. The battery specifications and experimental conditions are shown in [ 9]. Despite the fact that the two batteries have a nominal capacity that is similar, they differ in their advised operating currents, voltage restrictions, thermal tolerances, and efficiency. This offers an ideal setting for evaluating the generalizability of the suggested model. In connection with these samples, a collection of four comprehensive case studies was subsequently developed. The initial cost for the new battery is 530 SAR/kWh [ 25]. In contrast, the retired battery, which retains a 40% SoH, is valued at 75 SAR/kWh [ 25]. The efficiency specified for both charging and discharging processes is 0.95 [ 25 Two of these case studies employed low-stress circumstances, which were defined by a stable temperature, low C-rate, and moderate DoD, to establish baseline forecast accuracy. In the remaining scenarios, the ARDM’s capacity to respond to operational disruptions in real time was evaluated by creating high-stress operating conditions, such as deep DoD cycling, quick C-rate fluctuations, and heat variations. The model was evaluated across a broad spectrum of consumption scenarios, including those that replicate the highly irregular dispatch patterns that are common in grid-connected BESS that participate in tariff-driven arbitrage or frequency-regulation services, as a consequence of this stratified architecture. The experiments were carried out using a programmable cycling platform capable of imposing arbitrary current and voltage profiles on the test batteries. The hardware included a bidirectional programmable DC source/load, a battery management unit responsible for providing synchronized measurements of voltage, current, and SoC, and a precision temperature-control system capable of maintaining or varying the cell temperature between 20 °C and 45 °C. A high-resolution data acquisition (DAQ) system captured all operational variables at a sampling interval compatible with the ARDM’s real-time update cycle. A schematic of the testing architecture and the associated wiring layout is provided in , respectively, illustrating the controlled environment under which all degradation-relevant data were collected. Across all experimental scenarios, the batteries were cycled under profiles designed to emulate a wide range of realistic grid-storage conditions. The cycling patterns represented a broad spectrum of profiles, ranging from shallow DoD and low C-rate profiles modeling short-term grid-support operations to aggressive high- DoD and fast-charge/fast-discharge cycles modeling long-duration energy shifting. To evaluate the flexibility and sensitivity of the ARDM to thermal perturbations, the temperature was intentionally varied during high-stress runs (as opposed to the constant level during low-stress trials). This combination of stress factors was sufficient to demonstrate ARDM’s capability to predict degradation in steady and in quickly changing operational environments. Throughout each experiment, the ARDM operated continuously in parallel with the physical cycling process. Every 200 samples, the model recalculated its degradation parameters by minimizing the error between measured and predicted degradation increments. This real-time parameter-updating mechanism allowed ARDM to absorb new operational information, adjust its internal representation of degradation behavior, and forecast subsequent capacity fade with high precision, even when confronted with sudden changes in C-rate, DoD, or temperature. This versatility is crucial for real-world BESSs, since operational profiles frequently undergo quick changes in response to dynamic tariffs, changing renewable production, and system reliability requirements. The benchmark results produced by the experiment test the performance of the ARDM in a variety of scenarios. In Section 5, the model’s robustness, scalability, and flexibility for use in a self-learning digital-twin environment have been demonstrated. It provides excellent tracking accuracy for all scenarios and battery types. The trials thus verify ARDM as a viable and robust approach for the real-time degradation estimation in advanced energy-storage devices. 5. Results and Discussion The proposed system is evaluated under a DTM designed to enhance both the economic performance of BESS owners and the operational stability of the power system. Unlike conventional FTM, such as the one currently employed by the SEC, DTM creates time-dependent incentives that encourage fast, responsive participation from BESS operators. Under FTM, operators can charge during long off-peak intervals and discharge during extended on-peak windows without any temporal urgency. This tendency eventually restricts the contribution of storage assets to grid stability, lowers demand-side responsiveness, and reduces the flexibility of BESS scheduling. In contrast, DTM offers pricing signals that reflect real-time system circumstances, encouraging operators to alter charging and discharging operations in a manner that meets both grid demands and revenue maximization. A typical 5-MW/10-MWh BESS model is analyzed to estimate the economic effects of using DTM instead of FTM. This case study emphasizes the revenue disparities between the two tariff structures and offers regulatory authorities in Saudi Arabia and other locations that continue to depend on FTM procedures when developing market frameworks for storage participation with practical insights. 5.1. Input Data 5.1.1. Battery Specifications The estimated BESS system size is 5 MW/10 MWh. Two different batteries (Bat-1 and Bat-2) have been used to evaluate the impact of battery performance on the economic analysis of the BESS in the proposed study. The specifications of the used batteries are shown in [ 9]. The number of batteries for a 5 MW/10 MWh BESS is =10 MWH/2.4 kWh = 4170 units. 5.1.2. Tariff Mechanisms The FTM in Saudi Arabia is summarized in . The charging/discharging tariffs in FTM are shown in compared to the proposed DTM. The charging (off-peak) cost is 0.2 SAR/kWh; meanwhile, the discharging (on-peak) cost tariff is 0.35 SAR/kWh, with an average value of 0.275 SAR/kWh. The proposed dynamic tariff is selected from the tariff introduced in [ 30] with a similar average tariff as the one shown in the FTM (0.275 SAR/kWh). The tariff value represents the situation of the power system, where a low tariff means that there is surplus generation over the load demand, and vice versa. 5.1.3. Optimization Control Parameters 5.2. ARDM Results As has been shown in [ 9], the ARDM perfectly fits the real-time degradation of the batteries used in this study. The two batteries used in this study have been cycled at different C-rates, DoD, and temperatures. The variations in the cycling tests are shown in for Bat-1 and Bat-2, respectively. The ARDM is running every 200 samples to update the degradation model parameters for accurately calculating the degradation in the next 200 samples. The variation between the measured degradation and the calculated one from the ARDM at ( DoD = 0.9, C-rate = 0.5, and T = 25 °C) is shown in for Bat-1 and Bat-2, respectively. The ARDM’s better prediction accuracy was decisively shown by the quantitative findings of this rigorous validation process, in all cycling samples for the two batteries. The model demonstrated an exceptional degree of agreement between the anticipated and measured relative capacity values. The greatest variation across the scenarios was not greater than 0.04%, while the reported average prediction error consistently stayed below 0.005%. For example, the mean absolute change was an extraordinary 0.000513% in the low-stress condition (Bat-1). Additionally, the ARDM’s position was further cemented by comparison evaluations against three recognized benchmark degradation models, which demonstrated that it greatly outperformed traditional state-of-the-art techniques. The ARDM is a scalable, cost-effective, and accurate solution for degradation-aware energy management in both grid and electric vehicle applications, as the experimental data collectively indicate. 5.3. Optimization Algorithm Results As has been discussed above, the objective functions used in this study are very complex, and they need a very effective optimization algorithm to accurately get the optimal solution in a short time. For this reason, the MCA is selected due to its superior performance and has been examined in many applications [ 25, 26, 27, 28]. The MCA uses a high number of search agents in the initialization to enhance its exploration performance and gradually reduces its swarm size to enhance exploitation and reduce the convergence time. The comparison with the conventional algorithm is not fair without using the same idea. For this reason, the gradually reduced swarm size methodology has been applied to PSO and GWO in SRPSO and SRGWO [ 29, 30]. The control parameters of these optimization algorithms are summarized in . The number of search agents used in all optimization algorithms is 100, and the number of iterations is selected to be 30. The optimization is carried out for 50 individual runs to avoid the random nature associated with the performance of these optimization algorithms. The results obtained from these optimization algorithms used with ARDM and the SLDT are shown in , , respectively. These results showed that the MCA outperforms the other optimization algorithms in the accuracy of the fitness values and the convergence time. It is clear from this Table that the convergence time in ARDM for MCA is 4.15 s compared to 7.8 s and 6.5 s for SRPSO and SRGWO, respectively. Moreover, the convergence time for the OCL is 32.6 s, 54.7 s, and 49.5 s for the MCA, SRPSO, and SRGWO, respectively. These outstanding results of the MCA prove its superiority, and for this reason, it will be used in the coming studies. 5.4. Economic Models Results One day is selected to be used for comparison with an FTM and a DTM based on the data shown in Section 5.1. The FTM is charging the BESS during the off-peak period and discharging during the on-peak period for a single cycle a day. The SoC is starting with 10%, and its maximum value is 90% for both models for a fair compar
