Optimized configuration of a microgrid based on photovoltaics and hybrid energy storage considering battery degradation costs

Abstract

This paper proposes a capacity configuration method for a microgrid composed of a photovoltaic (PV) power generation system and a hybrid energy storage system (battery storage + supercapacitors). The core of this method involves constructing a mixed-integer linear programming (MILP) model and incorporating a battery aging model to determine the retirement time of the energy storage system, thereby optimizing the microgrid capacity configuration. Additionally, this paper explores the integration of supercapacitors (SCs) into the microgrid capacity configuration to effectively mitigate battery aging, enhancing the economic performance and operational efficiency of the microgrid. The proposed method is validated through real-world case studies, demonstrating its practical applicability in microgrid development. Moreover, the case study analysis compares the proposed method with traditional methods that do not consider battery aging factors and conducts a sensitivity analysis regarding changes in energy storage procurement cost parameters. The results demonstrate that the proposed method not only effectively addresses the challenges posed by battery aging but also offers superior economic and technical performance in microgrid optimization planning and capacity configuration.

Keywords

microgrid capacity configuration hybrid energy storage system battery aging supercapacitors integration mixed-integer linear programming

Introduction

In recent years, the global electrification rate has significantly improved with the widespread deployment of distributed renewable energy, energy storage, and microgrid technologies across various countries and regions. The decline in investment costs and increase in conversion efficiency have spurred investments in electrification projects based on renewable energy sources.^1,2 However, the construction and operation of these projects are closely linked to factors such as regional investment budgets, geographical location, climate conditions, and the policy environment of the installation site.^3,4 Non-interconnected areas (NIAs) are not connected to the national grid due to geographical remoteness or economic barriers. They face significant challenges in resource availability, economic constraints, technical and maintenance issues, environmental and regulatory constraints, as well as social and community engagement, all of which severely impact the reliability and economic viability of microgrid projects.⁵ These challenges include high initial and operational costs, limited local expertise, regulatory uncertainties, and the need for community acceptance, making it essential to develop tailored solutions to ensure the success and sustainability of microgrid initiatives in NIAs. Additionally, deploying microgrid projects in these NIAs is challenging due to the lack of detailed operational data from similar projects and historical meteorological data.⁶ Therefore, it is crucial to develop mathematical models that optimize the capacity of various distributed generation technologies within microgrids, considering the need for uninterrupted power supply to critical loads during independent operation, and to conduct thorough technical and economic feasibility pre-assessments based on the region’s economic, technical, and environmental conditions.

Numerous studies have focused on the optimal capacity configuration of small-scale microgrids incorporating photovoltaic (PV) and energy storage systems (ESS).^7–10 However, in the context of optimal capacity configuration for PV-battery microgrids, the following issues have received less attention: (1) Battery capacity degradation impacting system operation and replacement costs, with hybrid energy storage systems (HESS) used to extend battery life; (2) Variations in electricity prices, encompassing both main grid energy costs and costs due to critical load supply interruptions; and (3) Dependence on the main grid, including grid availability profiles. (1) Battery capacity degradation and its impact on system operation, which involves the replacement costs of battery storage and the use of hybrid energy storage systems (HESS) to extend battery life. (2) Variations in electricity prices, which should not only include the energy costs of supplying loads from the main grid but also account for the costs incurred by users due to interruptions in critical load supply. (3) Dependence on the main grid, including grid availability profiles. Although recent studies have incorporated battery energy storage system (BESS) aging models,^11,12 most do not consider the eventual need for BESS replacement, even though the lifecycle of battery storage is typically shorter than the average lifespan of PV modules. Reference 13 proposes a novel set of formulations for optimally sizing ESS in microgrids (MGs) by considering technical characteristics, service life, depth of discharge (DOD), and capacity degradation to minimize total scheduling costs and enhance economic feasibility and precision. Similarly, a novel BESS sizing model that accurately minimizes microgrid costs by incorporating major battery degradation factors is presented in Ref. 14, including temperature, DOD, cycles, state of charge, and time passage, into a linear framework for estimating BESS capacity loss. Reference 15 proposes a comprehensive optimization method for designing residential PV-battery microgrids, focusing on minimizing the levelized cost of energy (LCOE) while considering power supply loss limitations, operational constraints, and detailed battery lifetime estimation.

HESSs that combine supercapacitors with battery storage have garnered significant attention in recent studies evaluating the optimal scale of battery storage, due to their ability to more effectively manage stored energy and extend battery life.^16,17 Due to the benefits brought by HESS in frequency regulation services, system cost minimization, and microgrid energy management, they have seen applications at both the public grid level and in standalone DC microgrids.^18,19 As mentioned in Ref. 18, supercapattery devices, which combine the advantages of batteries and supercapacitors, have garnered attention for their high specific energy and power density, addressing the challenge of enhancing energy density without sacrificing power density. The integration of supercapacitors into microgrids offers significant economic and environmental benefits. Economically, supercapacitors can reduce the frequency of deep charge and discharge cycles experienced by batteries, thereby extending battery life and reducing replacement costs. Environmentally, by managing transient power fluctuations and stabilizing the state of charge (SOC) of batteries, supercapacitors help to improve the overall efficiency and reliability of the microgrid, leading to higher utilization of renewable energy sources and lower greenhouse gas emissions. These benefits make the integration of supercapacitors a compelling solution for enhancing the sustainability and economic viability of microgrids. Reference 20 presents a novel optimization approach that integrates system costs, SOC management for batteries and supercapacitors, to optimize the sizing and cost of a HESS in an independent Direct Current (DC) microgrid powered by solar photovoltaics, ensuring battery longevity and system stability. A two-level controller for a HESS combining lead-acid batteries and supercapacitors to optimize grid integration of photovoltaic plants in distribution grids is introduced in Ref. 19, providing peak power shaving and PV output power ramp limitation while minimizing battery degradation.

Given the importance of maintaining power supply continuity during outages, incorporating the cost of energy quality into the sizing methodology is crucial. References 21,22 quantified the economic losses during power outages by introducing the concept of interruption costs and the average load loss value (VOLL) based on customer surveys, emphasizing the necessity of detailed survey data for accurately assessing the cost of unserved energy (ENS). For example, Ref. 22 introduces a novel method for incorporating the value of resilience into the techno-economic optimization of PV and BESS, showing that including resilience considerations can increase system capacities and make previously uneconomical systems viable. For microgrid operation and energy management, Ref. 23 proposes a low-carbon economy operation strategy for multiple integrated energy microgrids (IEMs) using a double-layer Stackelberg game model, incorporating carbon trading and seasonal variations in PV data, to achieve collaborative low-carbon economic operation and multidimensional performance evaluation. Reference 24 constructs a wind-PV-hydrogen microgrid system and develops an optimal scheduling model that integrates long-term hydrogen storage and short-term battery storage to achieve economic optimization. An AI-based approach using the one-to-one based optimizer (OOBO) for economic dispatch and load management in three linked microgrids is proposed in Ref. 25, integrating renewable energy resources and optimizing scheduling to reduce operating costs and improve grid performance. For healthcare facilities, ensuring energy continuity is paramount with high ENS costs (CENS) and VOLL, especially in unstable grid environments in developing countries’ NIAs. Therefore, microgrid sizing strategies need to be flexibly adjusted to accommodate load fluctuations and grid uncertainties, ensuring power continuity while optimizing operational costs.

To address these unresolved issues, this paper proposes a microgrid capacity sizing method for systems comprising photovoltaic generation, battery storage, and supercapacitors. This method introduces an innovative approach by incorporating an ESS throughput aging model, which is designed to accurately estimate storage replacement costs. The model achieves this by calculating the additional energy required to compensate for the operational degradation that occurs over time. Additionally, the method integrates HESSs, utilizing supercapacitors to extend battery life. A photovoltaic performance model is incorporated into the sizing method to evaluate the techno-economic feasibility of various generation technologies and select the optimal solution. Grid intermittence is considered through availability profiles during the predicted microgrid operation period, and dynamic critical load interruption costs (CENS) are evaluated using adjusted load profiles to assess the dynamic criticality of the loads. Finally, the proposed microgrid capacity sizing method is applied to two case studies, further validating its effectiveness.

Basic models for PV and HESS

This section provides a detailed overview of the models utilized to evaluate key factors affecting microgrid performance and reliability. It covers the power output derating of photovoltaic (PV) systems due to varying ambient temperatures at installation sites, highlighting how higher temperatures can significantly reduce the efficiency and overall energy output of PV panels. This assessment ensures accurate projection of the PV system’s capacity under real-world conditions, leading to more reliable and optimized microgrid designs. Additionally, the section delves into the aging dynamics of HESSs, which combine batteries with supercapacitors. These systems are crucial for balancing supply and demand, providing stability, and enhancing the lifespan of the energy storage components. The aging dynamics model accounts for the gradual capacity degradation over time, essential for planning maintenance schedules and replacement strategies to ensure continuous and efficient operation.

PV power output derating model

The PV power output derating model is a mathematical model that takes into account various factors such as PV panel and ambient temperatures, irradiance, air quality, dust accumulation, and the actual efficiency and degradation of PV modules to predict and assess the reduction in power output of photovoltaic systems. By utilizing this model, we can accurately estimate the actual generating capacity of PV systems under different conditions, assess performance degradation over time, and optimize operation strategies to improve system efficiency and stability. The PV output derating model accounts for factors such as temperature, shading, and module degradation, which can affect the actual power output of photovoltaic panels. Specifically, the efficiency of PV cells is significantly influenced by ambient temperatures, with power output typically declining as temperatures rise.²⁶ Generally, the model uses the following equation to calculate the derated output power while approximating this effect²⁷:

P^{PV} (I_{T}, T) = P_{r}^{PV} \frac{I_{T}}{I^{STC}} (1 + γ^{PT} (T (I_{T}, v^{wind}, t^{amb}, k) - T^{STC}))

(1)

where P^PV(I_T,T) denotes the PV output corresponding to the irradiance level I_T and the operational temperature of the module

T (I_{T}, v^{wind}, t^{amb}, k)

P_{r}^{PV}

is the rated power of an individual PV panel under standard conditions and γ^PT represents the power temperature coefficient of the PV panel. I_STC and T_STC are the standard test condition values for irradiance and ambient temperature, respectively, set at 1000 W/m² and 25°C. Actually, to avoid overestimating photovoltaic (PV) output, a thermodynamic model that can correct the temperature of PV panels is proposed in Ref. 28. Therefore,

T (I_{T}, v^{wind}, t^{amb}, k)

indicates that the operational temperature of the module is mainly influenced by the irradiance intensity

I_{T}

, wind speed

v^{wind}

, ambient temperature

t^{amb}

, and other factors

k

. In order to more precisely evaluate the installed capacity of PV stations, the thermodynamic model is employed in this study for considering the influence of irradiance intensity and wind speed at the PV installation site on the temperature of PV modules.

HESS aging dynamics model

The Ah throughput model is an ideal tool for integrating battery aging effects into energy storage system (ESS) capacity planning.²⁸ Within our research framework, a battery is considered to have reached the end of its service life when its nominal energy capacity degrades to 80% of its initial value. Based on this criterion, we can calculate the total energy availability ( $E^{av}$ ) of a specific battery unit and subsequently evaluate the battery degradation rate per unit of discharge power. The calculation formula is as follows:

\deg^{ESS} = \frac{20 % E^{B}}{N C^{B} E^{B}} = \frac{E^{av}}{N C^{B} E^{B}}

(2)

where

E^{B}

denotes the rated energy capacity of the battery unit and

N C^{B}

means the maximum possible count of the charge/discharge cycles.

This study particularly focuses on the integration of hybrid energy storage systems (HESS) that combine supercapacitors and batteries in microgrid capacity planning. According to the Ah throughput model, incorporating OPzV supercapacitors with only 0.1% of the total energy capacity of the battery bank can significantly enhance battery lifespan by approximately 7.7%. This aging estimation model is highly versatile and flexible as it relies solely on parameters obtained through technical charge/discharge tests, making it applicable to all types of batteries.²⁸

Mathematical model

The primary goal of this paper is to propose a resource allocation method designed to identify the most cost-effective and technically feasible combination of photovoltaic (PV) panels, battery storage units, and supercapacitor (SC) units for configuring the capacity of a microgrid within a set investment budget, as shown in Figure 1. The proposed method achieves this by comparing various PV and energy storage system (ESS) units based on their technical and economic factors. The objective function considers both initial investment costs and long-term operational costs, ensuring a holistic economic analysis. Critical factors like load demand variations, meteorological conditions, and energy supply availability are incorporated to ensure a realistic assessment of microgrid performance and reliability, ultimately aiming to optimize these aspects along with cost-effectiveness. The end result is a robust and efficient energy management solution that guarantees the microgrid’s ability to reliably and economically meet consumer energy demands.

Figure 1.

Schematic diagram of the microgrid topology, including PV arrays, battery systems, supercapacitors, and charge controllers.

Objective function

The objective function of the proposed MILP model is a cost minimization function for optimizing microgrid (MG) sizing, incorporating various cost components related to photovoltaic (PV) installations, battery installations, grid interactions, backup power, and energy not supplied (ENS). The objective function is expressed as follows:

\min Z = \sum_{i \in Ω^{PV}} (C_{i}^{PV} + C_{i}^{PVinst}) X_{i}^{PV} + \sum_{j \in Ω^{B}} C_{j}^{B} X_{j}^{B} + \sum_{y = 1}^{L Y} \frac{1}{{(1 + r)}^{y}} (\sum_{j \in Ω^{B}} \frac{C_{j}^{B}}{E_{j}^{a v}} ε_{j}^{B} + \sum_{t \in T} (C_{t}^{grid} (P_{t}^{G} + \sum_{j \in Ω^{B}} P_{j, t}^{GB}) + C_{t}^{bckp} P_{t}^{D} + C_{t}^{ENS} P_{t}^{ENS} - C_{t}^{PVG} \sum_{i \in Ω^{PV}} P_{i, t}^{PVG}) Δ t)

(3)

In Ref. (3), Z is the total cost to be minimized. The first term in the first row sums up the costs associated with installing PV systems across all nodes i in set

Ω^{PV}

. Here,

C_{i}^{PV}

represents the capital cost of the i-tℎ PV system, and

C_{i}^{PVinst}

accounts for any installation-specific costs.

X_{i}^{PV}

is a binary variable indicating whether the i-tℎ PV system is installed (1) or not (0). The second term in the first row sums the costs of installing batteries across all members j in set

Ω^{B}

, where

C_{j}^{B}

is the cost of the j-tℎ battery, and

X_{j}^{B}

is a binary variable. The second row of the objective function is discounting future costs over the lifetime (LY) of the project. The discount rate is

{(1 + r)}^{- y}

, where r is the interest rate per period. Inside the discounted sum, the first term accounts for the replacement or decay cost of batteries over their lifetime, calculated by dividing the battery cost

C_{j}^{B}

by its average lifespan energy output

E_{j}^{a v}

and then multiplying by a factor

ε_{j}^{B}

, which represents degradation or efficiency loss. The following terms within the parentheses capture the costs associated with interacting with the grid over all time period t in set T. Specifically,

C_{t}^{grid} (P_{t}^{G} + \sum_{j \in Ω^{B}} P_{j, t}^{GB})

represents grid purchase cost, including both general grid purchases

P_{t}^{G}

and grid purchases specifically for charging batteries

P_{j, t}^{GB}

C_{t}^{grid}

is the grid purchase cost coefficient.

C_{t}^{bckp} P_{t}^{D}

calculate backup power cost.

C_{t}^{bckp}

is the backup power cost coefficient and

P_{t}^{D}

is the backup power.

C_{t}^{ENS} P_{t}^{ENS}

means penalty cost for energy not supplied to meet demand, calculated by the cost coefficient

C_{t}^{ENS}

and the unserved energy

P_{t}^{ENS}

- C_{t}^{PVG} \sum_{i \in Ω^{PV}} P_{i, t}^{PVG}

is revenue from selling PV-generated electricity back to the grid, calculated by the price coefficient

C_{t}^{PVG}

and the corresponding power

P_{i, t}^{PVG}

Constraints

1. Investment budget constraint

Constraint (4) represents a budget limitation in the optimization problem. It states that the total cost of installing photovoltaic (PV) systems across all nodes i plus the total cost of installing batteries across all j should not exceed a predefined maximum budget ( $C^{budget}$ ).

\sum_{i \in Ω^{PV}} (C_{i}^{PV} + C_{i}^{PVinst}) X_{i}^{PV} + \sum_{j \in Ω^{B}} C_{j}^{B} X_{j}^{B} \leq C^{budget}

(4)

2. Installation constraints of PV modules and batteries

Constraint (5) determines the installation status $X_{i}^{PV}$ of a PV system at node i. The floor function (⌊⌋) suggests that the number of PV systems that can be installed is an integer value. $N^{PVMax}$ represents the maximum volume or capacity of PV installations that can be considered strategic, and $V_{i}^{OCU}$ refers to the occupied volume or space at node i. Thus, constraint (5) calculates the maximum number of PV systems that can fit at a location based on space availability, and $X_{i}^{PV}$ inherits this installation decision from $X_{i}^{PVs}$ while also respecting the spatial constraint. Similarly, for batteries at location j, $X_{j}^{B}$ is determined by the binary decision variable $X_{j}^{Bb}$ for battery installation consideration in (6). $N^{BattMax}$ denotes the total maximum volume or capacity for batteries, and $V_{j}^{B}$ is the volume or space required for one battery at location j. This equation ensures that the number of batteries installed does not exceed what the location can physically accommodate. Constraint (7) sets a limit on the total area $A^{av}$ available for PV installations across all nodes. $A_{i}^{PV}$ represents the area required for installing a PV system at node i. The sum over all i ensures the total allocated area does not surpass the available area.

X_{i}^{PV} = X_{i}^{PVs} \cdot ⌊ N^{PVMax} / V_{i}^{OCU} ⌋, \forall i \in Ω^{PV}

(5)

X_{j}^{B} = X_{j}^{Bb} \cdot ⌊ N^{BattMax} / V_{j}^{B} ⌋, \forall j \in Ω^{B}

(6)

\sum_{i \in Ω^{PV}} A_{i}^{PV} X_{i}^{PV} \leq A^{av}

(7)

3. General power balance and PV generation balance constraints

Constraint (8) represents the overall power balance at each time period t. It enforces that the sum of the power from the main grid ( $P_{t}^{G}$ ), PV power to the load ( $P_{i, t}^{PVL}$ ), battery discharging power ( $P_{j, t}^{BL}$ ), backup power from diesel generators ( $P_{t}^{D}$ ), minus losses ( $P_{t}^{ENS}$ ), must equal the total load demand ( $P_{t}^{L}$ ) at any given time. Constraint (9) governs the operation of PV systems and their interaction with batteries at node i for each time period t. It ensures that the sum of PV power to the load ( $P_{i, t}^{PVL}$ ), PV power to the main grid ( $P_{i, t}^{PVG}$ ), sent to batteries ( $P_{i, j, t}^{PVB}$ ), and curtailed ( $P_{t}^{PVcurt}$ ) equals the theoretical maximum power production potential ( $X_{i}^{PV} P_{i, t}^{PVmpp}$ ) of the PV system at that node and time, adjusted for whether a PV system is installed.

P_{t}^{G} + \sum_{i \in Ω^{PV}} P_{i, t}^{PVL} + \sum_{j \in Ω^{B}} P_{j, t}^{BL} + P_{t}^{D} + P_{t}^{ENS} = P_{t}^{L}, \forall t \in T

(8)

P_{i, t}^{PVL} + P_{i, t}^{PVG} + \sum_{j \in Ω^{B}} P_{i, j, t}^{PVB} + P_{t}^{PVcurt} = X_{i}^{PV} P_{i, t}^{mpp}, \forall i \in Ω^{PV}, \forall t \in T

(9)

4. Energy transactions and backup-diesel-generator availability constraints

Constraint (10) sets a limit on the main grid power to load ( $P_{t}^{G}$ ) and main grid power to BESS ( $P_{j, t}^{GB}$ ) at any time t. In other words, the total power from the main grid cannot exceed maximum grid power available ( $P_{G}^{Max}$ ) and the availability of the grid at that time ( $G r i d_{t}^{av}$ ). Constraint (11) limits the total power generated by all PV systems ( $P_{i, t}^{PVG}$ ) across all nodes i in the network at time t. It ensures that the aggregate PV output does not surpass the maximum allowable PV generation capacity ( $P_{P V G}^{Max}$ ), adjusted by the grid’s availability. $B c k p_{t}^{av}$ denotes the backup diesel generator’s availability in (12), where $P_{D}^{Max}$ means the maximum power of the backup diesel generator.

P_{t}^{G} + \sum_{j \in Ω^{B}} P_{j, t}^{GB} \leq P_{G}^{Max} \cdot G r i d_{t}^{av}, \forall t \in T

(10)

\sum_{i \in Ω^{PV}} P_{i, t}^{PVG} \leq P_{P V G}^{Max} \cdot G r i d_{t}^{av}, \forall t \in T

(11)

P_{t}^{D} \leq P_{D}^{Max} \cdot B c k p_{t}^{av}, \forall t \in T

(12)

5. Battery operation constraints

Constraints (13)–(18) govern the operation of battery storage systems within the network, ensuring that charging and discharging operations respect the physical limitations of the batteries. For every time period t, these constraints ensure proper control of battery charging/discharging based on its state and capacity. It is noted that batteries can be charged from PV stations during periods of high photovoltaic generation, as well as from the grid. The two charging powers of batteries are represented by variables $P_{i, j, t}^{PVB}$ and $P_{j, t}^{GB}$ , respectively, and their charging states are also restricted by binary variables $B_{j, t}^{PVch}$ and $B_{j, t}^{Gch}$ , respectively. Constraint (13) indicates that the total power from PV stations should not exceed the upper limit defined by the battery’s usable capacity. Constraint (14) indicates that the total power should not exceed the maximum charging rate. Constraint (15) ensures that the total power should not be less than the maximum discharging rate in reverse. Constraint (16) ensures that the total power exchanged with the power grid is limited by their usable capacity. Constraint (17) indicates that the charging power from the grid is restricted by the battery’s maximum charging rate, controlled by the binary variable $B_{j, t}^{Gch}$ indicating charging status. Constraint (18) ensures that the discharge is controlled in a similar manner, allowing discharge when the battery is not scheduled for charging. In both cases, M is typically a large number ensuring that when the binary variable allows charging or discharging, the inequality becomes effectively an equality, reflecting the actual limits.

\sum_{i \in Ω^{PV}} P_{i, j, t}^{PVB} \leq X_{j}^{B} P_{j}^{Bu}, \forall j \in Ω^{B}, \forall t \in T

(13)

\sum_{i \in Ω^{PV}} P_{i, j, t}^{PVB} \leq M \cdot B_{j, t}^{PVch}, \forall j \in Ω^{B}, \forall t \in T

(14)

\sum_{i \in Ω^{PV}} P_{i, j, t}^{PVB} \geq - M \cdot (1 - B_{j, t}^{PVch}), \forall j \in Ω^{B}, \forall t \in T

(15)

P_{j, t}^{GB} \leq X_{j}^{B} P_{j}^{Bu}, \forall j \in Ω^{B}, \forall t \in T

(16)

P_{j, t}^{GB} \leq M \cdot B_{j, t}^{Gch}, \forall j \in Ω^{B}, \forall t \in T

(17)

P_{j, t}^{GB} \geq - M \cdot (1 - B_{j, t}^{Gch}), \forall j \in Ω^{B}, \forall t \in T

(18)

Similarly, constraints (19)–(21) presents three limits for the discharging power of the batteries. And constraint (22) enforces that at any given time period t, a battery j can only be in one of three states. $B_{j, t}^{dch}$ is a binary variable indicating the discharging status of the batteries.

P_{j, t}^{BL} \leq X_{j}^{B} P_{j}^{Bu}, \forall j \in Ω^{B}, \forall t \in T

(19)

P_{j, t}^{BL} \leq M \cdot B_{j, t}^{dch}, \forall j \in Ω^{B}, \forall t \in T

(20)

P_{j, t}^{BL} \geq - M \cdot (1 - B_{j, t}^{dch}), \forall j \in Ω^{B}, \forall t \in T

(21)

B_{j, t}^{PVch} + B_{j, t}^{Gch} + B_{j, t}^{dch} \leq 1, \forall j \in Ω^{B}, \forall t \in T

(22)

Constraints (23)–(25) define the state-of-charge (SoC) dynamics and constraints for batteries in an energy management system, particularly for optimizing the use of batteries in conjunction with renewable energy sources and grid interactions. The SoC changes over time is described in (23). Constraint (24) restricts the minimum and maximum SoC level. And constraint (25) defines the initial SoC level. $S D_{j}^{B}$ means the self-discharge rate. $η^{ch}$ and $η^{dch}$ represent the charge/discharge efficiencies, respectively. $E_{j}^{Bmin}$ is the minimum amount of energy and $E_{j, t}^{cap}$ represents the current capacity of the batteries. $E_{j}^{B}$ means the rated energy capacity of the battery unit. $S o C^{init}$ indicates the initial level coefficient of SoC.

S o C_{t, j} = S o C_{j, (t - 1)} (1 - S D_{j}^{B}) + (\sum_{i \in Ω^{PV}} P_{i, j, t}^{PVB} + P_{j, t}^{GB}) η^{ch} Δ t - P_{j, t}^{BL} Δ t / η^{dch}, \forall j \in Ω^{B}, \forall t \in T

(23)

X_{j}^{B} E_{j}^{Bmin} \leq S o C_{t, j} \leq B_{j, t}^{cap}, \forall j \in Ω^{B}, \forall t \in T

(24)

S o C_{j, t} = X_{j}^{B} E_{j}^{B} S o C^{init}, \forall j \in Ω^{B}, \forall t \in T

(25)

Constraints (26)–(28) extend the battery model to consider capacity degradation over time and enforce a limit on total degradation over the lifetime of the battery. Constraint (26) reiterates the initial battery capacity setup, where $B_{j, t_{0}}^{cap}$ is the capacity at the beginning of the analysis period (t0), determined by the battery existence binary and its nominal energy capacity. Constraint (27) describes the capacity degradation over time. Here, the capacity at each time step t is updated by subtracting the degradation effect caused by discharging the battery during that period. The rate of degradation is represented by $\deg_{j}^{ESS}$ , which could be a constant or a function accounting for factors such as cycling frequency, depth of discharge, and temperature. Constraint (28) ensures that the total reduction in battery capacity from the start (t0) to the end (t_f) of the analysis period does not exceed 20% of its initial capacity divided by the lifetime of the battery system (LY), plus a small tolerance $ε_{j}^{B}$ . This models a practical limit on how much a battery’s capacity can degrade over its operational lifetime, with the tolerance allowing for minor deviations. These constraints together form a comprehensive model of battery capacity management, including the effects of degradation over time and setting a limit to ensure the battery remains useful for its intended lifespan.

B_{j, t_{0}}^{cap} = X_{j}^{B} E_{j}^{B}, \forall j \in Ω^{B}

(26)

B_{j, t}^{cap} = B_{j, t - 1}^{cap} - \deg_{j}^{ESS} P_{j, t}^{BL}, \forall j \in Ω^{B}, \forall t \in T

(27)

B_{j, t_{0}}^{cap} - B_{j, t_{f}}^{cap} \leq 0.2 B_{j, t_{0}}^{cap} / L Y + ε_{j}^{B}, \forall j \in Ω^{B}

(28)

Finally, the proposed optimal resource allocation model can be summarized as follows:

\min Z

(29a)

s . t . InvCons = {(4) - (7)}

(29b)

OperCons = {(8) - (28)}

(29c)

Case study

In this section, the proposed MG sizing methodology is evaluated by applying two different cases studies.

Parameter setting

In the test cases considered in this paper, three distinct photovoltaic (PV) module technologies and two prevalent battery technologies were factored into the simulations. Additionally, the integration of supercapacitors (SCs) was evaluated for each technology and case study. Relevant parameters utilized in the simulations, are detailed in Tables 1 and 2.^29,30 Wind speed, solar irradiance, and temperature profiles employed in the test cases are illustrated in Figure 2. Additionally,

P_{G}^{Max}

is set as 1200 kW, and

P_{P V G}^{Max}

is 72 kW. The interest rate is set as 6%. The life year is 25 years.

C^{budget}

is 25,000$.

Table 1.

Parameters of candidate PV module technologies.

Parameter	Type
Parameter	Poly	Mono1	Mono2
$C_{i}^{PV}$ ($)	126	130	138
$C_{i}^{PVinst}$ ($)	220	220	220
$A^{av}$ (m²)	1.68	1.94	2.28
$P_{r}^{PV}$ (W)	345	375	390
$γ^{PT}$ (%/°C)	−0.35	−0.39	−0.37

*Poly, Mono1 and Mono2 mean three types of PV module technologies, i.e., Polycrystalline, Monocrystalline 1 and Monocrystalline 2, respectively.

Table 2.

Parameters of candidate storage technologies.

Parameter	Li-ion	VRLA	Li-ion + SC	VRLA + SC
$C_{j}^{B}$ ($)	1830	860	1837.12	864.99
$E^{B}$ (kWh)	3.6	2.4	3.6	2.4
$E_{j}^{Bmin}$ (kWh)	0.36	0.48	0.36	0.48
$B_{j, t}^{cap}$ (kWh)	3.6	2.4	3.6	2.4
$P_{j}^{Bu}$ (kW)	1.8	1.2	1.8	1.2
$η^{ch}$ / $η^{dch}$ (%)	0.95	0.9	0.95	0.9
$N C^{B}$	3000	1000	3231	1077
$S D_{j}^{B}$ (%/h)	0.0001075	0.000125	0.0001075	0.000125

Figure 2.

Climatic condition data including (a) wind speed, (b) solar irradiance, and (c) temperature.

Results for case 1

The input parameters for Case 1 are set as follows: $A^{av}$ is 3000 m², $P_{D}^{Max}$ is 15 kW, $C_{t}^{bckp}$ is 45 USD/kWh, and $C_{t}^{ENS}$ is 12 USD/kWh. Additionally, to reflect higher implicit costs during specific periods, such as from 10:00 to 14:00, the cost of ENS is adjusted by multiplying by a factor of 20 and 10, which forms two scenarios for comparison.

These scenarios, labeled as Case 1A and Case 1B, respectively, represent different levels of $C_{t}^{ENS}$ . In Case 1B, the higher $C_{t}^{ENS}$ leads to a greater objective function value, a smaller PV array size, and a larger ESS capacity, enhancing the solution’s robustness with increased storage capacity. Specifically, the PV generation in Case 1A comprises two sub-arrays with different technologies, while the ESS includes 6 Li-ion+SC units expected to remain throughout the project’s lifetime.

Comparing economic indicators reveals that with a higher

C_{t}^{ENS}

, the capacity configuration minimizes ENS during the operational horizon, improving reliability, but reduces energy exported to the grid, thereby reducing economic benefits. Consequently, the total cost of the microgrid project increases. Therefore, setting the value of

C_{t}^{ENS}

reasonably is crucial for planners to balance reliability and economic benefits, achieving a trade-off based on their preferences (Tables 3 and 4).

Table 3.

Capacity allocation results for Case 1.

	PV generators		ESS
Case	Type	Number	Type	Init number	Replace number
Case 1A	Mono2/Poly	18/24	Li-ion + SC	6	0
Case 1B	Mono2	56	VRLA + SC	5	1

Table 4.

Economic indices for Case 1.

Case	Case 1A	Case 1B
Investment for PV	10,788.00	20,048.00
Investment for ESS	11,022.72	4324.95
Cost for ESS maintenance	0.00	47.88
Cost for energy from grid	47,928.91	51,263.52
Cost for energy from backup generator	2819.97	4784.64
Cost for energy not supplied	0.00	14,845.60
Cost for energy sold to grid	−20,708.05	−47,522.91
Objective value	51,851.55	47,791.68

Results for Case 2

In Case 2, $C_{t}^{ENS}$ is set according to the load variations throughout the day. During peak load hours from 6:00 to 18:00, the ENS cost coefficient is the same as the marginal $C_{t}^{ENS}$ . However, during other off-peak periods, due to the significantly reduced economic losses from interrupted energy service, the ENS cost is set to one-tenth of the marginal cost.

In this case, we compare two scenarios: in Case 2A, the proposed capacity configuration model is utilized, while Case 2B assumes no degradation in the battery energy storage system (BESS). This means excluding the incorporation of supercapacitors (SC) in the HESS for extending battery life, not considering the linear battery degradation model, and omitting estimation of storage replacement units based on equipment degradation. Consequently, the objective function (5) excludes the term ( $\sum_{j \in Ω^{B}} \frac{C_{j}^{B}}{E_{j}^{a v}} ε_{j}^{B}$ ), constraints (24) and (25) are omitted, and constraint (23) is replaced with $B_{j, t}^{cap} = X_{j}^{B} E_{j}^{B}$ , thus not modeling battery degradation and replacement units in the capacity configuration.

The capacity configuration outcomes and economic metrics for both scenarios are presented in Tables 5 and 6, respectively. In Case 2A, the setup comprises PV modules of Mono2 and a HESS with 5 VRLA+SC units. No battery replacement is considered due to the relatively low ENS cost, allowing for the possibility of exporting energy to the main grid.

Table 5.

Capacity allocation results for Case 2.

	PV generators		ESS
Case	Type	Number	Type	Init number	Replace number
Case 2A	Mono2	64	VRLA + SC	5	0
Case 2B	Mono1	64	Li-ion/VRLA	1/5	0

Table 6.

Economic indices for Case 2.

Case	Case 2A	Case 2B
Investment for PV	22,912.00	22,400.00
Investment for ESS	4324.95	6130.00
Cost for ESS maintenance	0.00	0.00
Cost for energy from grid	35,144.75	25,693.01
Cost for energy from backup generator	0.00	0.00
Cost for energy not supplied	30,676.91	15,811.63
Cost for energy sold to grid	−42,934.61	−31,219.84
Objective value	50,123.99	38,814.80

Comparing Tables 5 and 6, in Case 2B, where battery aging and replacement are not considered, the number of storage units increases, and their technology composition changes, excluding SC coupling in the BESS. However, the increased BESS operation in Case 2B does not affect battery lifespan, resulting in SCs being excluded from the capacity configuration results. Neglecting BESS degradation underestimates the benefits of SC coupling, leading to higher operational costs when implementing the microgrid. Therefore, the exclusion of the BESS degradation model in Case 2B leads to an optimistic estimation of expected ENS and the microgrid’s interaction with the main grid, underestimating the benefits of SC coupling.

Comparing the outcomes in Tables 5 and 6 for Case 2B reveals a critical influence of neglecting battery aging and replacement on the design and operation of the battery energy storage system (BESS) in the microgrid. In Case 2B, where these factors are not considered, the number of storage units rises, and the technology mix shifts, particularly excluding the integration of supercapacitors (SCs) with the BESS. This oversight underestimates the advantages of incorporating SCs into the storage system, leading to higher operational costs in real-world implementation. In practice, SCs could potentially decrease battery replacement frequency and maintenance requirements, thus reducing overall operational expenditures. Additionally, disregarding BESS degradation in Case 2B provides an optimistic assessment of the expected energy not supplied (ENS) and the microgrid’s interaction with the main grid. This optimistic view fails to account for potential challenges arising from neglecting battery aging, such as increased unreliability and decreased energy availability during peak demand. Consequently, the true benefits of SC coupling in enhancing the microgrid’s performance, reliability, and cost-effectiveness are underestimated.

Sensitivity analysis

The variation in component costs plays a significant role in shaping the capacity configuration scheme. This section conducts a sensitivity analysis to examine how changes in the costs of energy storage systems (ESS), with both battery and supercapacitor (SC) costs reduced by the same percentage, affect the capacity configuration and economic indicators. The results of this analysis are summarized in Table 7.

Table 7.

Planning results and economic indices for different ESS investment costs.

	Case 1			Case 2
ESS investment cost	90%	80%	70%	90%	80%	70%
PV modules	Poly (44)	Poly (44)	Poly (44)	Mono2(64)	Mono2(64)	Mono1(64)
ESS units	Li-ion + SC(6)	Li-ion + SC(6)	Li-ion + SC(6)	VRLA + SC(6)	VRLA + SC(6)	Li-ion + SC(6)
Investment for PV	15,224.00	15,224.00	15,224.00	22,912.00	22,912.00	22,400.00
Investment for ESS	9920.45	8818.18	7715.90	4670.95	4151.95	3632.96
Cost for ESS maintenance	0.00	0.00	0.00	0.00	0.00	0.00
Cost for energy from grid	46,871.37	46,871.37	46,870.82	35,103.04	35,144.75	34,408.43
Cost for energy from backup generator	2735.88	2735.79	2736.98	0.00	0.00	0.00
Cost for energy not supplied	0.00	0.00	0.00	30,677.87	30,676.91	15,933.65
Cost for energy sold to grid	−21,917.41	−21,917.32	−21,916.95	−42,934.61	−42,934.61	−38,573.87
Objective value	52,834.29	51,732.02	50,630.75	50,429.24	49,950.99	37,801.17

From the findings presented in Table 7, it becomes clear that a decrease in ESS investment costs influences the configuration of storage units in Case 2 microgrid (MG), even though no battery replacements are required in either scenario. This suggests that while configuring battery storage assists in supporting load supply during specific periods, the primary focus remains on exporting energy to the grid, as indicated by the reduction in the objective function value.

Upon a thorough examination of the two scenarios, it becomes evident that the microgrid (MG) responds differently to reductions in energy storage system (ESS) unit prices. In Case 1, a decrease in ESS prices triggers a proportional increase in the number of photovoltaic (PV) modules, reaching a total of 48 additional modules as prices decline to 90%, 80%, and 70% of the baseline. Meanwhile, the number of ESS units remains static. In contrast, Case 2 exhibits a static PV module count despite ESS price reductions. However, when prices hit 70% of the original cost, a strategic shift occurs, introducing a new storage technology to enhance the MG’s energy management. The economic analysis underscores the significance of interrupted energy service costs in optimizing MG capacity, balancing project costs and minimizing unserved energy. The potential for energy export to the main grid also plays a crucial role, generating revenue to offset costs and influencing the MG’s capacity configuration and overall objectives. Restricted export prompts the MG to prioritize local load demands, fostering the use of storage units for reliability and resilience.

Conclusion

This paper presents a comprehensive methodology for optimizing the capacity configuration of a microgrid (MG) system, incorporating photovoltaics (PV), hybrid energy storage systems (HESS), and diesel generators. It considers various factors such as installation location, meteorological conditions impacting PV performance, and battery aging dynamics. The methodology introduces a novel approach to estimating storage replacement costs using the linear throughput model of the battery energy storage system (BESS), accounting for additional compensation energy required. Furthermore, it explores the advantages of integrating supercapacitors into HESS to extend battery life. Specific constraints and key factors, including the BESS degradation model, replacement calculation, HESS performance, main grid availability, and dynamic cost of interrupted energy service (CENS), are integrated into the decision-making process.

The proposed methodology’s effectiveness is validated through two case studies, showing its practical applicability. Comparative analysis with a methodology overlooking BESS degradation indicates inaccurate estimations of ENS and grid interaction. Neglecting BESS degradation costs can underestimate the benefits of using supercapacitors to enhance HESS performance, including efficiency gains, reliability improvements, and cost savings. This could lead to incorrect MG project capacity configurations and higher operating costs. Furthermore, sensitivity analysis based on energy storage system (ESS) costs reveals that reductions in ESS costs significantly influence MG capacity configuration, especially in scenarios with higher costs of interrupted energy service (CENS). Note that the proposed method is limited by reliance on historical weather and load data from specific regions. The computationally intensive MILP model may pose challenges for real-time optimization in large-scale microgrids. Future research directions include developing more efficient algorithms for real-time optimization, exploring the integration of multiple renewable energy sources to enhance reliability and economic viability, investigating advanced control strategies using machine learning and artificial intelligence, and conducting long-term performance analysis to assess sustainability and resilience over extended periods.

Footnotes

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by State Grid Hubei Electric Power Co., Ltd. Major Science and Technology Project: Research on key technologies for active distribution network structure optimization and operation coordination adapting to the friendly interaction of microgrid clusters (project code: 521532220008).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Gielen

Boshell

Saygin

, et al. The role of renewable energy in the global energy transformation. Energy Strategy Rev 2019; 24: 38–50.

Nadeem

Siddiqui

Khalid

, et al. Distributed energy systems: a review of classification, technologies, applications, and policies. Energy Strategy Rev 2023; 48: 101096.

Hamawandy

Wahid

, et al. Renewable energy resources investment and green finance: evidence from China. Resour Policy 2021; 74: 102402.

Khezri

Mahmoudi

Aki

. Optimal planning of solar photovoltaic and battery storage systems for grid-connected residential sector: review, challenges and new perspectives. Renew Sustain Energy Rev 2022; 153: 111763.

Uddin

Dong

, et al. Microgrids: a review, outstanding issues and future trends. Energy Strategy Rev 2023; 49: 101127.

Twaisan

Barışçı

. Integrated distributed energy resources (DER) and microgrids: modeling and optimization of DERs. Electronics 2022; 11(18): 2816.

Mathew

Hossain

Saha

, et al. Sizing approaches for solar photovoltaic-based microgrids: a comprehensive review. IET Energy Syst Integration 2022; 4(1): 1–27.

León

Romero-Quete

Merchán

, et al. Optimal design of PV and hybrid storage based microgrids for healthcare and government facilities connected to highly intermittent utility grids. Appl Energy 2023; 335: 120709.

Korjani

Casu

Damiano

, et al. An online energy management tool for sizing integrated PV-BESS systems for residential prosumers. Appl Energy 2022; 313: 118765.

10.

Hossain

Shareef

Hossain

, et al. Hybrid PV and battery system sizing for commercial buildings in Malaysia: a case study of FKE-2 building in UTeM. IEEE Trans Ind Appl 2024; 60(3): 4933–4945.

11.

Kazhamiaka

Rosenberg

Keshav

. Tractable lithium-ion storage models for optimizing energy systems. Energy Inform 2019; 2(1): 4.

12.

Yuan

Sorknæs

Lund

, et al. The bidding strategies of large-scale battery storage in 100% renewable smart energy systems. Appl Energy 2022; 326: 119960.

13.

Amini

Khorsandi

Vahidi

, et al. Optimal sizing of battery energy storage in a microgrid considering capacity degradation and replacement year. Elec Power Syst Res 2021; 195: 107170.

14.

Amini

Sanjareh

Nazari

, et al. A novel model for battery optimal sizing in microgrid planning considering battery capacity degradation process and thermal impact. IEEE Trans Sustain Energy 2024; 15(3): 1435–1449.

15.

Alramlawi

. Design optimization of a residential PV-battery microgrid with a detailed battery lifetime estimation model. IEEE Trans Ind Appl 2020; 56(2): 2020–2030.

16.

Jayasawal

Karna

Thapa

. Topologies for interfacing supercapacitor and battery in hybrid electric vehicle applications: an overview. In: 2021 International Conference on Sustainable Energy and Future Electric Transportation (SEFET), Hyderabad, India. 2021. p. 1–6.

17.

Tang

Ding

, et al. Research on control strategy of hybrid energy source system based on interconnection and damping assignment. In: 2023 IEEE 6th International Electrical and Energy Conference (CIEEC), Hefei, China. 2023. p. 704–709.

18.

Iqbal

Supercapattery

. Merging of battery-supercapacitor electrodes for hybrid energy storage devices. J Energy Storage 2022; 46: 103823.

19.

Díaz-González

Chillón-Antón

Llonch-Masachs

, et al. A hybrid energy storage solution based on supercapacitors and batteries for the grid integration of utility scale photovoltaic plants. J Energy Storage 2022; 51: 104446.

20.

Premadasa

PND

Chandima

. An innovative approach of optimizing size and cost of hybrid energy storage system with state of charge regulation for stand-alone direct current microgrids. J Energy Storage 2020; 32: 101703.

21.

Marqusee

Becker

Ericson

. Resilience and economics of microgrids with PV, battery storage, and networked diesel generators. Adv Appl Energy 2021; 3: 100049.

22.

Laws

Anderson

DiOrio

, et al. Impacts of valuing resilience on cost-optimal PV and storage systems for commercial buildings. Renew Energy 2018; 127: 896–909.

23.

Zhang

Wang

, et al. Carbon-oriented optimal operation strategy based on Stackelberg game for multiple integrated energy microgrids. Elec Power Syst Res 2023; 224: 109778.

24.

Shaker

Keshta

Mosa

, et al. Energy management system for multi-interconnected microgrids during grid connected and autonomous operation modes considering load management. Sci Rep 2024; 14(1): 24359.

25.

Qiu

, et al. Optimal scheduling for microgrids considering long-term and short-term energy storage. J Energy Storage 2024; 93: 112137.

26.

Wandji Nyamsi

Lindfors

. Detecting clear-sky periods from photovoltaic power measurements. Meteorol Appl 2024; 31(3): e2201.

27.

Kim

, et al. Study on actual Ah-throughput-based health indicator of battery module consisting of inconsistent cells for its second life. Electron Lett 2022; 58(15): 588–590.

28.

King

Boyson

Kratochvil

. Photovoltaic array performance model . Albuquerque, NM: Sandia National Laboratories; 2004 (SAND-2004-3535).

29.

Gaetani-Liseo

Alonso

Jammes

. Impacts of supercapacitors on battery lifetime in hybrid energy storage system in building integrated photovoltaic DC micro-grid. In: 2018 7th International Conference on Renewable Energy Research and Applications (ICRERA), Paris, France. 2018. p. 1247–1252.

30.

Liu

, et al. Performance assessment of different photovoltaic module technologies in floating photovoltaic power plants under waters environment. Renew Energy 2024; 222: 119890.