Integration Of Storage Systems Research Articles

Energy savings by adapting consumer behavior in grid-connected photovoltaic systems with battery storage

Today, the world faces many energy challenges that make the use of clean energy an obligation and an emergency. These challenges require changes in a wide range of sectors, including transport, industry and residential areas. At present, energy production is responsible for a large amount of greenhouse gas emissions, the main cause of climate change with its various dangerous effects on human life. Therefore, a change towards non-carbon energy is becoming a necessity. Certainly, this evolution has been underway for years and the renewable energy market has developed in a surprisingly efficient way. However, the current situation has reached an alarming state and requires the active participation of all stakeholders, especially consumers. Therefore, the main objective of this paper is to illustrate how consumer behavior can significantly influence and contribute to the optimization of renewable energy systems, especially photovoltaic systems. The paper emphasizes the beneficial integration of batteries and storage systems to achieve energy savings. The results show that with some adjustments in daily behavior, an overall energy saving of 62% can be achieved compared to the normal consumption scenario: the energy obtained from the grid and then the electricity bill is reduced.

Assessment of carbon dioxide transcritical cycles for electrothermal energy storage with geological storage in salt cavities

Carbon capture, utilisation, and storage (CCUS) technologies are envisaged as critical actors in the energy transition and climate change mitigation framework. Despite the recent advances in CCUS implementation, there is still significant scope for their growth and improvement. On the other hand, developing new large-scale energy storage systems is a key factor for the massive deployment of renewable energy systems. Technologies such as the electrothermal energy storage system based on carbon dioxide transcritical cycles incorporating geological storage (CEEGS) can contribute to both fields. Preliminary studies have shown that the integrated storage system can operate with roundtrip efficiencies exceeding 50%, injecting over 1 million tons of captured carbon dioxide (CO2) annually. It is an early-stage technology with open challenges for successful development, such as an adequate balance between the high- and low-temperature thermal storage reservoirs or the operation definition between the electrothermal system and the carbon dioxide injection and recovery. Underground pressures will oscillate with the charge/discharge process, with operational implications and constraints. Advancing in the concept requires the definition of optimised operation. This work presents a novel analysis of the adaptation of CO2 transcritical cycles to include injection and production processes with the specific scenario of salt cavities. It considers new modifications to adapt the transcritical cycles in closed-mode operation to the conditions required by the salt cavities injection and production processes. Different case studies are evaluated, evaluating the impact of injection and geological storage, production and surface storage, and reinjection into the geological formation. The analyses show roundtrip efficiencies within the range of 49.1–73.0% when the injection, production, and reinjection processes are executed.

A bidirectional isolated DC‐DC converter with low voltage stresses for energy storage systems integration

A bidirectional isolated DC‐DC converter with low voltage stresses for energy storage systems integration

Design, analysis, and operation optimisation of a shipborne absorption refrigeration–ice thermal storage system based on waste-heat utilization

The integration of waste-driven absorption refrigeration technology into marine applications is recognized as a promising low-carbon solution. Nonetheless, its practical application does not inherently outperform electric compression technology, primarily due to the absorption system’s inherent thermal inertia and limited adjustability. This characteristic renders the system less responsive to the dynamic refrigeration requirements typically encountered on board vessels. In this study, a shipborne ammonia absorption refrigeration–ice storage system is proposed to balance the cooling load between supply and demand. To evaluate the efficiency of the integrated system for ice storage and cooling during ship sailing, key operational variables including refrigeration temperature, cooling water temperature, and generation temperature are analyzed to assess the thermal cycle performance of the system. A typical shipping route is considered for cooling operation optimisation of the absorption refrigeration–ice storage system. This paper explores four typical cooling-supply strategies and determines the operation modes for the ice storage period and the cooling period based on considerations of energy consumption, energy utilization, and economic factors. The final optimisation results indicate that the combined system has significant advantages based on energy consumption and operation cost. In comparison to compression and absorption refrigeration systems, the proposed system exhibits 52.4% and 8.31% reductions in energy consumption, respectively. Additionally, the operation costs are decreased by 12.27% and 27.41%, respectively, and the life cycle costs are lowered by 11.51% and 32.35%, respectively. The findings of this combined system can provide a technical reference for the design and operation of a ship waste-heat refrigeration system.

Inventory model having preservation technology with fix lifetime under two level trade credit policy

Supply-chain management involves moving storage supplies from origin to consumption, with manufacturers running production based on quadratic demand, distributors and retailers monitoring inventory. When a new product is released, demand often rises linearly and then declines dramatically when an alternative becomes available. Shortages are not allowed. Players' inventory will decrease at a rate of (1/(1+m-t)), where m is fixed lifetime, greater than the replenishment time. Deteriorating goods experience constant mass loss or usefulness, but preservation technology can help the damaged item to be consumed. Retailers with direct customer relationships can reduce stock spoilage through good warehouses. Manufacturers' storage systems have a higher deterioration rate. Two-tier trade credit financing is examined in this model. Distributors offer specific credit terms to stores, while manufacturers provide a grace period for invoicing. Distributors and retailers must pay interest on unsold inventories if invoices aren't settled on time. An integrated storage system reduces costs by minimizing costs through multiple shipments from manufacturers to distributors and retailers, and by adjusting replenishment times for each player. The resolution process is designed so that the supply chain operator gets the best possible decision. Therefore, results are authorized using mathematical examples for different scenarios. Management decisions are suggested.

Optimised linear active disturbance rejection control of multiport-isolated DC-DC converter for hydrogen energy storage system integration

Hydrogen energy storage systems are becoming increasingly accepted owing to their environmental friendliness. The efficiency and performance of these systems largely depend on the attributes of their power electronic interface systems. Among the promising solutions is a multiport-isolated DC-DC converter with characteristics of reduced component count, fewer conversion stages, and galvanic isolation. However, this system presents a challenge due to its inherent cross-coupling effect, complicating precise control. To address this, linear active disturbance rejection control (LADRC) is a viable option, leveraging dynamic/disturbance properties observed by linear extended state observers. LADRC serves as a decoupling controller, mitigating the cross-coupling effect. However, LADRC has several gains, and selecting them can be a difficult task, often requiring manual tuning. To streamline this process, this paper proposes utilising particle swarm optimisation to determine the optimum gains of LADRC. By employing this approach, the implementation of LADRC is facilitated with reduced design efforts while ensuring effective decoupling control within the system. Simulations are conducted to validate the performance of the optimised gain LADRC.

The safety and environmental impacts of battery storage systems in renewable energy

The integration of battery storage systems in renewable energy infrastructure has garnered significant attention due to its potential to enhance energy reliability, efficiency, and sustainability. However, alongside these benefits, concerns persist regarding the safety and environmental impacts associated with the deployment and operation of such systems. This review explores the multifaceted aspects of safety and environmental considerations in battery storage systems within the context of renewable energy. Firstly, safety concerns encompass a range of factors, including thermal runaway, fire hazards, and chemical leakage, which pose risks to both human life and property. Mitigation strategies such as advanced battery management systems and fire suppression technologies are critical for addressing these risks effectively. Secondly, environmental impacts arise throughout the lifecycle of battery storage systems, from raw material extraction to end-of-life disposal. Key issues include resource depletion, greenhouse gas emissions, and pollution from mining activities. Sustainable practices such as responsible sourcing of materials, recycling initiatives, and the development of second-life applications are essential for minimizing environmental footprints. Furthermore, the interaction between battery storage systems and renewable energy sources introduces complexities in assessing environmental impacts. While battery storage facilitates the integration of intermittent renewables like solar and wind by providing grid stabilization and energy storage capabilities, its environmental benefits may be compromised by factors such as energy-intensive manufacturing processes and reliance on non-renewable resources. The safety and environmental impacts of battery storage systems in renewable energy demand comprehensive evaluation and management strategies to maximize benefits while minimizing risks. Collaboration among stakeholders, technological innovation, and regulatory frameworks are crucial for promoting the sustainable deployment and operation of these systems in the transition towards a cleaner and more resilient energy future.

Enhancing grid stability through predictive control and fuzzy neural networks in flywheel energy storage systems integration

Renewable energy systems, exemplified by solar and wind power, are increasingly integrated into modern power grids to mitigate environmental impact and reduce reliance on fossil fuels. However, the inherent intermittency and unpredictability of renewable energy sources pose challenges to grid stability and reliability. Energy Storage Systems (ESS) offer a promising solution to address these challenges by smoothing out power fluctuations and ensuring a consistent power supply. Among various ESS technologies, Flywheel Energy Storage Systems (FESS) have emerged as a noteworthy contender due to their rapid response times, low operating costs, and extended lifespan. This paper focuses on investigating the operation of a novel unit comprising a solar power system integrated with a Flywheel Energy Storage System (PV-FESS). The aim is to develop an effective control algorithm utilizing adaptive fuzzy neural networks and model predictive control (ANFIS-MPC) to manage power fluctuations stemming from renewable energy sources within the grid. The proposed control strategy aims to optimize the operation of the PV-FESS system by dynamically adjusting the energy absorption or release of the flywheel to maintain grid stability. Simulation studies conducted using Matlab-Simulink/Simcape software validate the efficacy of the ANFIS-MPC algorithm in mitigating abnormal fluctuations from renewable energy sources. The results demonstrate that the PV-FESS system effectively balances power fluctuations, ensuring a stable and reliable power output to the grid.

Enhanced water production in hemispherical solar stills via solar collector and thermal storage integration: experimental validation

ABSTRACT In arid regions that face water scarcity, solar distillation offers hope by meeting the increasing need for clean drinking water. This study investigates the integration of a hot water storage system, heated by a flat plate solar collector, into a hemispherical solar still. Placing the storage tank below the still's absorber enhances heat input and efficiently stores excess daytime thermal energy. Conducted in Bouismail from December 2022 to October 2023, the study yielded significant results: the modified solar still outperformed the conventional one, with production increases of 157% in winter and 207% in summer. Moreover, the modified still demonstrated remarkable efficiency improvement in summer, reaching 37.42% compared to 20.38% for the conventional still. The orientation of the fins within the storage tank, with respect to the hot water entrance orifice, significantly impacted water production, with alterations of the angle resulting in decreases of up to 25%. Increasing saltwater depth led to reduced distilled water output, with declines of up to 37.08% for the modified still. Economic analysis showed a slightly higher cost per liter of water produced by the modified still ($0.1897) compared to the simple one ($0.1446).

Research Progress and Application Prospects of Solid-State Hydrogen Storage Technology.

Solid-state hydrogen storage technology has emerged as a disruptive solution to the "last mile" challenge in large-scale hydrogen energy applications, garnering significant global research attention. This paper systematically reviews the Chinese research progress in solid-state hydrogen storage material systems, thermodynamic mechanisms, and system integration. It also quantitatively assesses the market potential of solid-state hydrogen storage across four major application scenarios: on-board hydrogen storage, hydrogen refueling stations, backup power supplies, and power grid peak shaving. Furthermore, it analyzes the bottlenecks and challenges in industrialization related to key materials, testing standards, and innovation platforms. While acknowledging that the cost and performance of solid-state hydrogen storage are not yet fully competitive, the paper highlights its unique advantages of high safety, energy density, and potentially lower costs, showing promise in new energy vehicles and distributed energy fields. Breakthroughs in new hydrogen storage materials like magnesium-based and vanadium-based materials, coupled with improved standards, specifications, and innovation mechanisms, are expected to propel solid-state hydrogen storage into a mainstream technology within 10-15 years, with a market scale exceeding USD 14.3 billion. To accelerate the leapfrog development of China's solid-state hydrogen storage industry, increased investment in basic research, focused efforts on key core technologies, and streamlining the industry chain from materials to systems are recommended. This includes addressing challenges in passenger vehicles, commercial vehicles, and hydrogen refueling stations, and building a collaborative innovation ecosystem involving government, industry, academia, research, finance, and intermediary entities to support the achievement of carbon peak and neutrality goals and foster a clean, low-carbon, safe, and efficient modern energy system.

A novel multi-port high-gain bidirectional DC–DC converter for energy storage system integration with DC microgrids

Bidirectional converters have often been used in numerous applications like DC microgrids, renewable energy, hybrid energy storage systems, electric vehicles, etc. The paper proposes a novel multi-port high-gain (NMPHG) bidirectional DC–DC converter that supports DC microgrid (DC-MG) applications. The main contributions of the proposed converter are high step-up/step-down conversion gain, multiple input ports, lower switch voltage stress, and lower component count owing to the single converter with multiple input ports for DC microgrid applications. The detailed operational principle, analysis, and design considerations of proposed NMPHG bidirectional DC–DC converters are discussed. Furthermore, the loss analysis, detailed comparison with similar works, and efficiency analysis with non-modalities during forward power flow (LV to HV) and reverse power flow (HV to LV) modes are presented. The efficiency of the proposed converter is found to be 93.8% in forward power flow and 92.9% in reverse power flow modes at rated power. Finally, a hardware prototype of the proposed NMPHG bidirectional DC–DC converters is implemented with 100 W in FPF mode and 200 W in RPF mode with a TMS320F28335 processor and validated with theoretical counterparts.

Integration and performance analysis of optimal large-scale hybrid PV and pump hydro storage system based upon floating PV for practical application

The widespread use of green energy sources creates a significant demand for energy storage. Hybrid floating photovoltaic (FPV) and pumped hydro storage (PHS) represent one of the most dependable and cost-effective solutions, which uses the PV system on the water body combined with a pair of lakes with different heights. This study focuses on the load side as well as the PHS capacity factor and aims to lower the cost of energy (COE) by raising the PHS capacity factor. Since building a PHS requires a significant upfront investment, doing so will also lower the COE and enable the acquisition of PHS for large-scale power production, which will guarantee power access in large cities. To simulate the FPV-PHS system, the multi-objective genetic algorithm (MOGA) is employed. Sufficient power management is a prerequisite for achieving system reliability in the best possible hybrid energy system design and implementation. Considering the 60-year system lifespan, the net present cost (NPC) analysis shows that, out of all the communities evaluated, the FPV-PHS system has the lowest NPC and COE. For the optimal configuration (FPV (Block A) 105 MW, PHS 80 MW, FPV (Block B) 357 MW, and the current hydropower plant), the expected NPC and energy costs for implementing the hybrid energy system (HRS) at the chosen location are $44,737,613 and $40/MWh, respectively. The existing FPV system spans 4.65 km2 and lowers the evaporation fraction by 17,279,400 m3. The hybrid FPV-PHS system reduces annual CO2 emissions by 581,830 tons. Our research reveals that a hybrid floating PV and pump storage hydropower system offers more steady clean electricity, implying a great significance for power grid infrastructure.

Integrating Compressed CO2 Energy Storage in an Integrated Energy System

The integration of an energy storage system into an integrated energy system (IES) enhances renewable energy penetration while catering to diverse energy loads. In previous studies, the adoption of a battery energy storage (BES) system posed challenges related to installation capacity and capacity loss, impacting the technical and economic performance of the IES. To overcome these challenges, this study introduces a novel design incorporating a compressed CO2 energy storage (CCES) system into an IES. This integration mitigates the capacity loss issues associated with BES systems and offers advantages for configuring large-scale IESs. A mixed integer linear programming problem was formulated to optimize the configuration and operation of the IES. With an energy storage capacity of 267 MWh, the IES integrated with a CCES (IES–CCES) system incurred an investment cost of MUSD 161.9, slightly higher by MUSD 0.5 compared to the IES integrated with a BES (IES–BES) system. When not considering the capacity loss of the BES system, the annual operation cost of the IES–BES system was 0.5 MUSD lower than that of the IES–CCES system, amounting to MUSD 766.6. However, considering the capacity loss of the BES system, this study reveals that the operation cost of the IES–BES system surpassed that of the IES–CCES system beyond the sixth year. Over the 30-year lifespan of the IES, the total cost of the IES–CCES system was MUSD 4.4 lower than the minimum total cost of the IES–BES system.

A multilevel optimization approach for daily scheduling of combined heat and power units with integrated electrical and thermal storage

Renowned for their remarkable overall efficiencies ranging from 70% to 90%, combined heat and power systems stand as a pivotal strategy for optimizing energy consumption by capitalizing on the synergistic relationship between electricity and thermal energy production. However, achieving optimal performance in combined heat and power systems remains a formidable challenge due to the intricate interplay of numerous variables. This paper presents a novel approach to daily scheduling optimization for combined heat and power units, focusing on the integration of electrical and thermal storage systems and meticulous consideration of security constraints. The optimization of combined heat and power unit scheduling introduces a mixed-integer nonlinear programming challenge, replete with deterministic and random variables. Addressing this complexity requires resilient solutions. In this study, we employ a multilevel optimization technique, transforming the problem into a bilevel structure. The initial step involves mapping operating parameters and minimizing costs through a water cycle optimization algorithm, laying a robust foundation for combined heat and power unit operation. Subsequently, we immerse ourselves in the realm of stochastic contingency scenarios, acknowledging the myriad uncertainties inherent in real-world systems. To empirically validate the efficacy of our proposed algorithm, extensive simulations are conducted on IEEE 18-bus and 24-bus test systems that emulate practical power networks. The results unequivocally showcase the power of our approach in navigating the complexities of optimal CHP unit planning. This paper's contributions lie in its innovative multilevel optimization technique, adeptly addressing both deterministic and stochastic aspects, ultimately paving the way for increased energy efficiency and enhanced system reliability in practical applications.

Design and implementation of a reduced switch seventeen-level multilevel inverter for grid integration of battery storage system

Multilevel inverters (MLI) have a variety of applications in battery storage system, renewable energy integration, electric vehicle (EV), drives for industry and in various flexible AC transmission system (FACTS). Applications of MLI in battery storage system have greatly increased over the past few decades. Conventional MLIs used for grid integration of battery energy storage is having large number of components and it requires transformer for grid integration. The large number of components and transformer increases the cost, size, weight and losses of the MLI, thus reduces the efficiency of the system. Keeping this in view, the goal of the proposed work is to design a novel reduced switch seventeen-level MLI which is having a lower number of components and do not require transformer for grid integration. This modular nature of proposed MLI is used to increase power handling capacity without introducing additional converters. For this, first proposed reduced switch seventeen-level MLI has been compared with other existing seventeen level topologies which in order to show the effectiveness of proposed topology. After analysis its effectiveness, the proposed MLI has been designed and then closed loop d-q current controller for grid connected mode is analyzed. The different curves are plotted to justify the stability of proposed inverter. The control of output voltage is also included using modified sine pulse width modulation (SPWM) method. The proposed MLI has been implemented in MATLAB/Simulink and validated using real time hardware-in-loop (HIL) emulator.

Integrated System of Mechanical Regulator and Electrical Circuitry on Triboelectric Energy Harvesting with Near‐Field Communication for Low Power Consumption

AbstractThis paper presents an integrated power generation, conversion, and storage system with a temperature monitoring system, including passive near‐field communication (PGCS‐TMS‐pNFC), based on triboelectric nanogenerators (TENGs). The system consists of three parts; first, a mechanical regulation system (MRS) with kinematic design and multiphase control; second, a phase control rectifying circuit (PCRC) with a battery charging circuit (BCC); and third, a data processing and communication system (TMS‐pNFC) for low power consumption. The MRS, consisting of a reel spring and gear train, achieves a high‐frequency output exceeding 1000 Hz, and long‐lasting operation. PCRC provides stable and minimizing fluctuated DC output with a ripple effect as RMS voltage and current of 423 V and 46 µA, for 2 s, with a crest factor of 1.15 and 1.08. The PGCS‐TMS‐pNFC, harvesting building energy, such as opening and closing doors, is used to sense and store temperature data for more than 29 min, and the data is transmitted to the receiver via pNFC through smartphone tagging. Additionally, the PGCS with BCC is implemented on a bicycle frame to charge electronic devices while riding. This research underscores the integrated approach and practical applications of TENG technology that contribute to energy efficiency and sustainability in the lives.

Temperature-Energy Relationships and Spatial Distribution Analysis for Nano-Enhanced Phase Change Materials Via Thermal Energy Storage

Despite the abundance and affordability of solar energy, its adoption in industrial and domestic sectors, especially in developing countries, still needs to be improved. This study addresses the gap by proposing integrated storage systems to align energy supply and demand, essential for various industrial processes. Investigating Nano-enhanced Phase Change Material (PCM), the research formulates governing equations for the phase change process, explores numerical simulations using MATLAB's Finite Volume Method, and validates models. The PCM comprises a solid salt mixture with Sodium Chloride Nanoparticles. The analysis of nano-enhanced PCMs for thermal energy storage focuses on understanding the interrelationship between temperature, energy, and nanoparticle distribution within the PCM. Visuals based on 3D surface plots and scatter plots illustrate how energy storage characteristics vary with temperature and spatial variables, identifying phase change temperatures and energy absorption/release points. These visualizations guide PCM optimization for improved thermal conductivity and heat capacity, which is crucial for diverse applications like solar energy systems and thermal management in electronics. Nano-enhanced PCM performance can be further enhanced by employing advanced numerical methods and simulation tools for accurate prediction and optimization before experimental validation.

Blockchain Opportunities for Water Resources Management: A Comprehensive Review

Blockchain technology has been used for the digitalization of physical asset management and data management processes in many areas in the industry and academia, including the water domain. Its potential as an immutable data storage system and smart contract integration has provided a plethora of use cases and utility in the domain of hydrology and water resources. This systematic review critically examines the application of blockchain technology in the field of water resources and hydrology. By analyzing 104 academic publications and 37 non-academic studies from 2017 to 15 October 2022, this paper identifies the current state of blockchain applications in water management, delineates their potential use cases, and assesses their practical utility and scalability. Despite the theoretical promise of blockchain for enhancing water governance, data security, and stakeholder trust, the review reveals a noticeable gap between theoretical potential and the existence of workable, real-time applications specifically in water resources management. The findings indicate that while blockchain technology has been effectively implemented in various sectors, its adoption in hydrological domain is still emerging, with limited empirical evidence to support full-scale deployment. The paper concludes with a call for more empirical research to validate theoretical benefits, address scalability and interoperability challenges, and integrate blockchain technology with real-time data networks for sustainable water management practices.

A Hybrid Energy Storage System Integrated with a Wave Energy Converter: Data-Driven Stochastic Power Management for Output Power Smoothing

Beyond solar and wind energy, wave energy is gaining great interest due to its very high theoretical potential, although its stochastic nature causes intermittent and fluctuating power production. Energy storage system (ESS) integration to wave energy converter (WEC) plants represents a promising solution to mitigate this issue. To overcome the technological limits of the single storage devices, the hybridization of complementary ESSs represents an effective solution, extending the operating range over different timeframes. This paper analyzes the benefits of Li-ion battery–supercapacitor hybrid ESS integration into a grid-connected WEC, aiming at smoothing the produced power oscillations. The hybridization concept involves coupling a power-intensive technology, such as a supercapacitor devoted to managing fluctuations at higher frequency, with a battery technology exploited to manage power variations over longer timeframes to mitigate degradation issues. In this study, a multi-objective data-driven power management strategy, based on the simultaneous perturbation stochastic approximation (SPSA) algorithm, is implemented to minimize power fluctuations in terms of power ramp (representing the power variation between two consecutive values with a 1 s time step), both at the Point of Common Coupling (PCC) and the Li-ion battery terminals, thanks to the supercapacitor peak-shaving function. The SPSA management strategy, together with a suitable sizing procedure, allows a reduction of more than 70% in the power oscillations at the PCC with respect to those at the WEC terminals, while decreasing battery stress by more than 25% if compared to a non-hybrid ESS consisting of a Li-ion battery. This shows how supercapacitor features can extend battery lifespan when integrated in a hybrid ESS.

Incorporating a Load-Shifting Algorithm for Optimal Energy Storage Capacity Design in Smart Homes

In today’s power system landscape, renewable energy (RE) resources play a pivotal role, particularly within the residential sector. Despite the significance of these resources, the intermittent nature of RE resources, influenced by variable weather conditions, poses challenges to their reliability as energy resources. Addressing this challenge, the integration of an energy storage system (ESS) emerges as a viable solution, enabling the storage of surplus energy during peak-generation periods and subsequent release during shortages. One of the great challenges of ESSs is how to design ESSs efficiently. This paper focuses on a distributed power-flow system within a smart home environment, comprising uncontrollable power generators, uncontrollable loads, and multiple energy storage units. To address the challenge of minimizing energy loss in ESSs, this paper proposes a novel approach, called energy-efficient storage capacity with loss reduction (SCALE) scheme, that combines multiple-load power-flow assignment with a load-shifting algorithm to minimize energy loss and determine the optimal energy storage capacity. The optimization problem for optimal energy storage capacity is formalized using linear programming techniques. To validate the proposed scheme, real experimental data from a smart home environment during winter and summer seasons are employed. The results demonstrate the efficacy of the proposed algorithm in significantly reducing energy loss, particularly under winter conditions, and determining optimal energy storage capacity, with reductions of up to 11.4% in energy loss and up to 62.1% in optimal energy storage capacity.