Two-Stage Multi-Objective Optimal Planning of Hybrid AC/DC Microgrid by Using ϵ-Constraint Method

Abstract

In this paper, a multi-objective mixed integer linear programming (MOMILP) approach is proposed for the optimal planning of battery energy storage systems (BESSs) and the interlink converter (ILC) in hybrid AC/DC microgrids (HMGs). The ILC is the backbone of the HMG, facilitating power exchange between the sub-grids. It plays a vital role in enhancing the stability of the HMG by balancing power between subsystems. Economically, the ILC enables the transfer of surplus power and lower-cost energy between the AC and DC microgrids. Therefore, selecting an optimal size for the ILC is critical from both technical and economic perspectives. However, existing studies have overlooked the optimal sizing of the ILC and its associated stress factors in the planning of HMGs. This paper proposes a multi-objective planning approach for HMGs that considers both calendar and cyclic ageing of BESSs. The performance of the proposed strategy is compared with the most widely used existing methods. The results confirm the superiority of the proposed planning approach in terms of both technical performance and economic efficiency.

1. Introduction

With the significant rise in DC generation resources, such as solar photovoltaic (PV) systems and battery energy storage systems (BESSs), along with the increasing prevalence of DC loads, including LED lighting systems and artificial intelligence data centres, DC microgrids (DCMGs) have become more economical due to the reduced need for multiple successive AC-DC power conversions. However, the use of AC microgrids (ACMGs) remains essential since large industrial and community-scale motors are predominantly AC induction motors, which are more cost-effective than DC motors [1]. Therefore, emerging hybrid AC/DC microgrids (HMGs) combine the benefits of both AC and DC networks simultaneously. An HMG comprises AC and DC generation units, AC and DC loads, and one or more parallel interlinking AC/DC converters (ILCs) that interconnect the ACMGs and DCMGs. The ILC plays a vital role in ensuring both the stability and economic operation of the HMG. It facilitates power-sharing between sub-grids, enabling flexible and smooth operation while enhancing economic efficiency by transferring surplus low-cost power to sub-grids experiencing high load conditions or expensive energy prices. Therefore, having an optimal size of the ILC is imperative for the optimal operation of the HMG, and it should be considered in the planning stage. In addition, the capacity fade of the BESS, along with the stress and degradation of the ILC, which contains semiconductor switches, should be considered for the optimal planning and operation of the HMG [2]. The capacity fade of the BESS is categorised into calendar ageing and cyclic ageing. Cyclic ageing is influenced by factors such as the depth of discharge (DoD) and the state of charge (SoC), while calendar ageing is associated with the storage conditions of the BESS over time. Consequently, in the operation stage, only cyclic ageing is typically considered, whereas, for planning and sizing problems, both types of capacity fade should be accounted for for [3]. The minimisation of microgrid planning costs, along with carbon emission reduction through the adoption of the Black Widow Optimisation algorithm, was discussed in [4]. A centralised adaptive energy management system based on fuzzy logic was proposed in [5] to optimise the operational costs of an islanded microgrid. A decentralised power flow model for a multi-area power system was proposed in [6] to ensure the smooth and uninterrupted operation of microgrids.

While the planning of ACMGs and DCMGs has been extensively explored in the literature [7,8], studies focusing on the planning of HMGs remain limited. The optimal planning of HMGs with a focus on $C{O}_2$ emission reduction was discussed in [9]. Although one of the objectives was the planning and management of BESSs, the study did not consider the calendar and cyclic ageing of the battery. Similarly, the planning of an HMG was proposed in [10], where various battery types were modelled to identify the most suitable option for the system’s operation. In Ref. [11], a comprehensive review was conducted on the planning and operation of HMGs; however, it did not highlight the gaps or future directions regarding the significance of optimal sizing and the degradation of ILCs. In Ref. [12], a constant value was used to model converter ageing in the planning of an HMG. However, the study did not consider the optimal sizing of the converter or account for the degradation of the BESS. References [13,14] proposed simplified planning methods for HMGs, neglecting the degradation costs and constraints associated with the BESS and ILC. In a related study on multi-microgrid systems (MMs) [15], a metaheuristic method was employed to address the planning problem. However, the degradation of the BESS and ILC were not taken into account, and the sizing of the ILC and tie-line between sub-grids was not optimised.

1.1. Research Motivation and Research Gaps

Although the literature on the planning of HMGs is limited, the existing works have not optimised the size of the ILC despite its critical role in the configuration and operation of the HMG. Furthermore, the degradation and ageing of the ILC, a semiconductor-based device, have not been adequately modelled or studied. This paper thoroughly addresses both aspects by incorporating them into the proposed planning framework. The proposed planning strategy is scalable to larger HMGs and offers valuable insights into achieving a fully renewable and zero-carbon emission electricity network.

1.2. Research Contributions

The contributions of this paper are as follows:

• The sizing of the ILC is modelled and integrated into the objective function to determine its optimal capacity, recognising its role as a pivotal component of HMGs.
• The degradation of the ILC, which affects the operation of HMGs, is modelled using the Arrhenius equation and fatigue analysis techniques commonly applied to semiconductor devices.
• The optimal planning problem is formulated within the framework of multi-objective optimisation, where the $ϵ$-Constraint method is employed to model and solve the objective functions.
• A reliability index based on energy not supplied (ENS) is proposed and integrated as an objective function to ensure that the planning method and its results adhere to the reliability standards set by the Australian Energy Market Commission.
• The energy throughput concept is implemented as a constraint to mitigate the cyclic ageing of the BESS. Additionally, the calendar ageing of the battery is modelled using an empirical approach.

The rest of the paper is organised as follows. Section 2 discusses the detailed methodology of the proposed planning system. Section 3 explains the $ϵ$-Constraint method, which is employed as a multi-objective optimisation approach in this study. Section 4 presents the case study along with its specifications. The planning results and a comprehensive discussion of the findings are provided in Section 5. Finally, Section 6 concludes the paper by summarising the key outcomes and contributions.

2. Methodology

2.1. Battery Degradation Modelling

Battery degradation is typically categorised into calendar ageing and cycle ageing. Calendar ageing is influenced by the battery’s storage conditions, while cycle ageing is determined by its usage patterns. Therefore, both calendar and cycle ageing effects should be accounted for in the optimal planning of HMGs employing BESSs [16]. An electro-thermal coupling model was proposed in [17,18] for the degradation modelling of lithium-ion batteries.

2.1.1. Cyclic Ageing: Energy Throughput Constraint

A battery’s energy throughput indicates its capacity to sustain a limited number of charge and discharge cycles, progressively reducing the amount of power it can deliver over time, as formulated in (1). Energy throughput can be represented as an operational constraint or incorporated into the economic scheduling of microgrid systems through a cost function [2]. In this paper, battery degradation related to energy throughput is modelled as a constraint, as formulated in (2).

$$E{\Pi }_{t,i}=E_i.Do{D}_{t,i}.{\eta }_i.E_r$$

(1)

The cyclic degradation constraint is as follows:

$$i\in \{AC,DC\}:\sum _{t=1}^T(P_{t,i}^{Chr}{\eta }_i\Delta t-{\displaystyle \frac{P_{t,i}^{Dis}\Delta t}{{\eta }_i}})\le E{\Pi }_{t,i}$$

(2)

where i denotes the battery that installed on the AC sub-grid or DC sub-grid. $E{\Pi }_{t,i}$ is the energy throughput of the battery at time t. $Do{D}_{t,i}$, ${\eta }_i$, and $E_r$ represent the depth of discharge, round-trip efficiency, and energy retention of the batteries, respectively. $P_{t,i}^{Dis}$ and $P_{t,i}^{Chr}$ are the instantaneous discharging and charging power of the batteries. $\Delta t$ shows the operation time interval of the HMG, which is considered 30 min.

2.1.2. Calendar Ageing

Several factors, including temperature, SoC level, and the instantaneous energy level of the BESS, influence the calendar ageing of a BESS. By assuming that the temperature is maintained constant through an appropriate cooling and ventilation system, the effect of temperature can be disregarded. Therefore, the linear calendar ageing function can be calculated as shown in (3). This paper adopts an empirical model that was previously investigated in [19].

$$\begin{array}{c}\hfill \{\begin{array}{c}\forall t\in T\\ i\in \{AC,DC\}\end{array}:\end{array}$$

$$CLAG(t)=[{({\displaystyle \frac{t}{720}})}^{0.8}-{({\displaystyle \frac{t-1}{720}})}^{0.8}](0.0028So{C}_i(t)+0.001939E_i(t))$$

(3)

2.2. ILC Stress Modelling

Transmitting power through semiconductor devices raises the junction temperature, which consequently induces fatigue or stress on the device, ultimately reducing its lifespan [20]. Considering this stress level and minimising it in the optimal planning and scheduling of HMG is crucial. The stress index, derived from the acceleration factor, is presented in (4)–(6). The acceleration factor, widely used for calculating semiconductor fatigue and failure probability, quantifies the impact of temperature-induced degradation [21].

$$P_{\Lambda }(t)=P_{ILC}(t).{\displaystyle \frac{1-{{\rm Y}}_{ILC}}{{{\rm Y}}_{ILC}}}$$

(4)

$$T_J(t)=P_{\Lambda }(t).R_{\theta }+T_a$$

(5)

$$\Xi (t)=e^{T_J(t).{\displaystyle \frac{k_B}{Ea}}}$$

(6)

where $P_{\Lambda }$ is the dissipated power of ILC and ${{\rm Y}}_{ILC}$ is the efficiency of the ILC. $T_J(t)$ is the junction temperature and has a linear relationship with the dissipated power of ILC [21]. $T_a$, $k_B$, and $E_a$ are the ambient temperature, Boltzmann constant ($8.617\times {10}^{-5}$ electronVolt (eV)/Kelvin(K)), and the thermal activation energy of the semiconductor (0.4 eV), respectively. $R_{\theta }$ is the junction-to-case thermal resistance (0.006 K/W) [22]. Equation (6) is a nonlinear function. To linearise it into a standard linear form, the natural logarithm (ln) is applied, and an auxiliary variable ($\Omega$) is introduced to represent the linearised expression of the stress index.

$$\Omega (t)=(P_{\Lambda }(t).R_{\theta }+T_a).{\displaystyle \frac{k_B}{Ea}}$$

(7)

2.3. Reliability Modelling

Energy not supplied is a reliability index that quantifies the amount of energy not delivered to the consumers due to generation curtailment or generation outage.This index is measured in megawatt-hours (MWh) as formulated in (8).

$$i\in \{AC,DC\}:ENS{(t)}_i=\sum _{k=1}^Q{P}_{k,i}^{NotSupplied}\times \Delta t$$

(8)

where k and Q represent the outage (not supplied) events and the total number of outages within each time interval, respectively. $P_{k,i}^{NotSupplied}$ defines the amount of load not supplied during each time interval.

2.4. System Constraints

To ensure that each component of the grid-connected HMG is planned and operates within its permissible limits, several constraints are defined as follows:

2.4.1. BESS Constraints

The energy level of the BESS and its charging/discharging power should remain within the specified maximum and minimum limits. Additionally, the BESS can charge or discharge at any given time but not simultaneously. These constraints are formulated in (9)–(14).

$$\begin{array}{c}\hfill \{\begin{array}{c}\forall t\in T\\ i\in \{AC,DC\}\end{array}:\end{array}$$

$$P_{t,i}^{BESS}=P_{t,i}^{Chr}-P_{t,i}^{Dis}$$

(9)

$$P_i^{Chr}(t)\le P_i^{BESS}{\psi }_i^{Chr}(t){\Lambda }_{Chr,i}^{Max}$$

(10)

$$P_i^{Dis}(t)\le P_i^{BESS}{\psi }_i^{Dis}(t){\Lambda }_{Dis,i}^{Max}$$

(11)

$${\psi }_i^{Chr}(t)+{\psi }_i^{Dis}(t)\le 1$$

(12)

$$0\le {\Lambda }_{Chr,i}^{Max},{\Lambda }_{Dis,i}^{Max}\le 1$$

(13)

$$\overline{{\Theta }_i}{E}_i^{BESS}\le E_i^{BESS}(t)\le E_i^{BESS}\underline{{\Theta }_i}$$

(14)

where ${\Lambda }_{Chr,i}^{Max}$, ${\Lambda }_{Dis,i}^{Max}$, $\overline{{\Theta }_i}$, and $\underline{{\Theta }_i}$ represent the maximum and minimum limits of the battery’s charging/discharging power and energy at each time interval, respectively.

Equations (10) and (11) are nonlinear and cannot be solved using the CPLEX solver version 22.1.1. Therefore, the Big-M method is employed to linearise them.

$$0\le {\zeta }_i(t)\le M{\psi }_i^{Chr}(t)$$

(15)

$$0\le {\kappa }_i(t)\le M{\psi }_i^{Dis}(t)$$

(16)

$${\zeta }_i(t)=P_i^{BESS}-{\kappa }_i(t)$$

(17)

$$P_i^{Chr}(t)\le {\zeta }_i(t)$$

(18)

$$P_i^{Dis}(t)\le P_i^{BESS}-{\zeta }_i(t)$$

(19)

where ${\zeta }_i(t)$ and ${\kappa }_i(t)$ are continuous auxiliary variables for the charging and discharging mode, respectively. M: is a large enough constant to handle the constraints.

2.4.2. ILC Constraints

The power-sharing capability between the ACMG and DCMG is bound by the IL’s maximum capacity.

$$\forall t\in T:P_t^{AC\leftrightarrow DC}=P_{t,\omega }^{AC\to DC}+P_t^{DC\to AC}$$

(20)

$$\forall t\in T:0\le |P_t^{AC\to DC}|\le P_{ILC}^{Max}{\text{Ϝ}}_t^{AC\to DC}$$

(21)

$$\forall t\in T:0\le |P_t^{DC\to AC}|\le P_{ILC}^{Max}{\text{Ϝ}}_t^{DC\to AC}$$

(22)

$$\forall t\in T:{\text{Ϝ}}_t^{AC\to DC}+{\text{Ϝ}}_t^{DC\to AC}\le 1$$

(23)

${\text{Ϝ}}_t^{AC\to DC}$ and ${\text{Ϝ}}_t^{DC\to AC}$ are binary variables that determine the direction of power-sharing between sub-grids. At each instance, power can be transferred either from the ACMG to the DCMG or vice versa.

Another constraint for the ILC is that its capacity must be sufficient to exchange power between sub-grids to facilitate the charging of the BESS.

$$P_{ILC}\ge Max\{P_{AC}^{BESS},P_{DC}^{BESS}\}$$

(24)

2.4.3. Upstream Grid Constraint

As previously discussed, the system is a grid-connected HMG capable of bidirectional energy exchange with the upstream grid. Therefore, the limitations on permissible power exchange must be observed during both the planning and operational stages. These constraints are formulated in (25) and (26).

$$\forall t\in T:P_t^{grid}=P_t^{Buy}-P_t^{Sell}$$

(25)

$$\forall t\in T:P_{Grid}^{Min}\le P_t^{grid}\le P_{Grid}^{Max}$$

(26)

2.4.4. Reliability Constraints

Two constraints are defined for the ENS reliability index. The first, modelled in Equation (27), specifies that 20% of the HMG’s load comprises critical emergency loads that cannot be shed. The second constraint, mandated by the Australian Energy Market Commission, requires that the expected unserved energy (ENS) must not exceed 0.002% of the total annual energy demand [23].

$$i\in \{AC,DC\}:ENS{(t)}_i\le 0.8.P_i^{Load}(t)\times \Delta t$$

(27)

$$\sum _{t=1}^{T=365}(EN{S}^{AC}(t)+EN{S}^{DC}(t))\le 0.002\%.\sum _{t=1}^{T=365}(P_{AC}^{Load}(t)+P_{DC}^{Load}(t))$$

(28)

2.5. Power Balance

The power balance of the HMG is separated into the power balance for the ACMG and the power balance for the DCMG. These two subsystems then exchange power via the ILC, as indicated in (29)–(31).

$$P_t^{grid}-P_{t,AC}^{BESS}+P_{t,AC}^{PV}+P_t^{AC\leftrightarrow DC}+\sum _{k=1}^Q{P}_{t,k}^{NotSupplied,AC}=\sum _{n=1}^N{P}_{t,n}^{Load,AC}$$

(29)

$$P_{t,DC}^{PV}-P_{t,DC}^{BESS}+P_t^{DC\leftrightarrow AC}+\sum _{k=1}^Q{P}_{t,k}^{NotSupplied,DC}=\sum _{m=1}^M{P}_{t,m}^{Load,DC}$$

(30)

2.6. Objective Functions

This work aims to determine the optimal capacity and power rating for both the AC-side and DC-side BESSs, along with the capacity of the ILC, to minimise the planning and operational costs of the BESS while maximising system reliability. This optimisation considers the degradation of both the BESSs and the ILC.

$$ObjectiveFunction=Min\{IC+OC,CLA{G}^{AC}+CLA{G}^{DC},\Omega ,EN{S}^{AC}+EN{S}^{DC}\}$$

(31)

where IC and OC represent the investment and operational costs of the HMG, respectively.

$$IC=CR{F}_{BESS}\times {\Phi }^{BESS}+CR{F}_{ILC}\times {\Phi }^{ILC}$$

(32)

The Capital Recovery Factor (CRF) is employed to annualise the capital costs of the BESSs and ILC, enabling the integration of operational and investment costs. This approach is necessary as the planning horizon or equipment lifespan typically ranges from 10 to 20 years, whereas operational costs are usually evaluated over a much shorter period. The mathematical expressions for the CRF and the investment costs of the BESS and ILC are formulated in (33)–(35).

$$CRF={\displaystyle \frac{IR{(1+IR)}^H}{IR{(1+IR)}^H-1}}$$

(33)

$$i\in \{AC,DC\}:{\Phi }_i^{BESS}=P_i^{BESS}.{\varphi }_p^{BESS}+E_i^{BESS}.{\varphi }_e^{BESS}$$

(34)

$${\Phi }^{ILC}=P^{ILC}.{\varphi }_p^{ILC}$$

(35)

where $IR$ represents the interest rate, set at 4.35% based on the Reserve Bank of Australia’s 2025 data, and H denotes the equipment lifespan, considered 20 years for the BESS and 10 years for the ILC. ${\varphi }_p^{BESS}$ and ${\varphi }_e^{BESS}$ are the unit power cost and unit capacity cost of the BESS, and their corresponding values are 250 AUD/kW and 600 AUD/kWh, respectively [24]. ${\varphi }_p^{ILC}$ is the ILC unit capacity cost and its value is 190 AUD/kW [25]. $E^{BESS}$, $P^{BESS}$, and $P^{ILC}$ are decision variables.

The operational costs include bidirectional energy transactions with the upstream grid for load supply and the charging and discharging of the BESSs, as formulated in Equation (36).

$$OC=\sum _{t=1}^T(P^{grid}(t)+P_i^{BESS}(t))\times {\varphi }^{energy}(t)$$

(36)

where ${\varphi }^{energy}(t)$ is the real energy price at each settlement period, which is obtained from the Australian National Electricity Market dashboard [26].

3. Multi-Objective Optimisation Algorithm

Since there is no conflict among the minimisation of investment costs, operational costs, calendar ageing of the BESS, and the degradation levels of the ILC, these objectives are addressed using the weighted sum model. However, a conflict arises between ENS reduction and the optimal sizing of the BESS and ILC, as increasing their size can lower ENS and improve system reliability but also results in higher planning costs. Therefore, the $ϵ$-Constraint multi-objective optimisation method is applied to solve the objectives as it allows for a more precise exploration of the trade-off between minimising ENS and planning costs by optimising one objective while imposing a constraint on the other, effectively avoiding the bias associated with weight selection in the weighted sum approach.

Finally, the objective functions can be modelled as below:

$$OF=Min\{O{F}_1,O{F}_2\}$$

(37)

$$O{F}_1={\Theta }_1.(IC+OC)+{\Theta }_2.(CLA{G}^{AC}+CLA{G}^{DC})+{\Theta }_3.\Omega$$

(38)

$$O{F}_2=ENS$$

(39)

Since cost is the dominant factor, followed by the calendar ageing of the battery due to its higher expense compared to the ILC, the weighting factors ($\Theta$) are assigned as 0.8, 0.15, and 0.05, respectively.

The procedure of $ϵ$-Constraint is explained in Algorithm 1 [27].

Algorithm 1 $ϵ$-Constraint Multi-Objective Method

1: function Define $O{F}_1$, $O{F}_2$ & constraints   2:      $O{F}_2$≤$ϵ$   3:      while $O{F}_1$= Deactive() do   4:            Max($O{F}_1$)   5:            Min($O{F}_2$)   6:      end while   7:      while $O{F}_2$= Deactive() do   8:            Min($O{F}_1$)   9:            Max($O{F}_2$) 10:      end while 11:      Create Pay-off Matrix 12:      Define Pareto Front Solution Numbers ($N_{Steps}$) 13:      Solve Model 14:      return Results (Pareto Solutions) 15:  end function

The planning optimisation is formulated as a multi-objective mixed-integer linear programming (MOMILP) problem using the Pyomo package version 6.7.3 in Python, with the CPLEX solver employed to solve the optimisation. The flowchart of the proposed planning optimisation is depicted in Figure 1.

As shown in Figure 1, the first stage uses the $ϵ$-Constraint method to determine the nominal capacity and power rating of the BESSs and ILC. These values are then used as inputs for the second optimisation stage, modelled as a MILP that aims to minimise operational costs.

4. Case Study and HMG Parameters

The single-line diagram of the HMG under study and its communication networks is shown in Figure 2. The specifications of the ACMG and DCMG are tabulated in

Table 1. AC sub-grid parameters.

Description	Notation	Value
Nominal capacity of the AC-side PV	$P_{n,AC}^{PV}$	500 kW
Minimum State of Charge of the BESS	$So{C}_{Min}^{BESS,AC}$	20%
Maximum State of Charge of the BESS	$So{C}_{Max}^{BESS,AC}$	90%
Round trip efficiency of the BESS	${\eta }^{Ch}={\eta }^{Dis}$	0.9

and

Table 2. DC sub-grid parameters.

Description	Notation	Value
Nominal capacity of the DC-side PV	$P_{n,DC}^{PV}$	300 kW
Minimum State of Charge of the BESS	$So{C}_{Min}^{BESS,DC}$	20%
Maximum State of Charge of the BESS	$So{C}_{Max}^{BESS,DC}$	90%
Round trip efficiency of the BESS	${\eta }^{Ch}={\eta }^{Dis}$	0.9

.

Real data on energy prices, PV generation, and loads were obtained from the Australian National Electricity Market and Griffith University.

5. Results and Discussion

Three study scenarios were modelled to assess the performance of the proposed strategy. Subsequently, the efficiency of the proposed planning method was compared with existing approaches to validate its superiority.

5.1. Scenario 1

In this scenario, the results of the proposed planning method are presented in

Table 3. Multi-objective planning results.

${\mathit{OF}}_1(\mathbf{AUD})$	${\mathit{OF}}_2(\mathbf{kW})$	${\mathit{E}}_{\mathit{AC}}^{\mathit{BESS}}(\mathbf{kWh})$	${\mathit{E}}_{\mathit{DC}}^{\mathit{BESS}}(\mathbf{kWh})$	${\mathit{P}}_{\mathit{AC}}^{\mathit{BESS}}(\mathbf{kW})$	${\mathit{P}}_{\mathit{AC}}^{\mathit{BESS}}(\mathbf{kW})$	${\mathit{P}}^{\mathit{ILC}}$
246,907.10	0.00	7022.66	6902.20	408.93	320.52	485.47
246,638.22	0.08	6960.36	6992.62	427.37	359.97	503.91
246,532.67	0.17	7105.59	6986.52	257.54	320.06	416.30
246,581.30	0.25	7078.37	6988.82	282.47	321.28	413.38
246,442.42	0.33	7019.05	6998.63	285.28	358.75	451.36
246,478.01	0.41	7289.26	6776.67	277.52	320.95	412.26
246,430.34	0.50	7211.96	6910.40	259.99	287.43	383.68
246,567.71	0.58	7043.81	6976.15	285.03	360.62	451.93
246,628.16	0.66	7081.39	6980.49	285.06	320.72	412.03
247,441.35	0.75	7077.57	6962.50	272.36	358.98	453.27
246,053.09	0.83	7106.13	6897.19	407.91	288.51	484.45

and illustrated in Figure 3 and Figure 4.

In Table 3, eleven rows of results are presented. This is because the problem was solved using a multi-objective optimisation method, which generates a Pareto front, offering flexibility for the system operator to select the most suitable solution from the available options. The eleven result rows correspond to the number of optimisation stages considered in the $ϵ$-Constraint method.

Figure 3 and Figure 4 illustrate the operational results of the DC and AC subsystems, respectively. To ensure clarity and simplify the presentation of the results, data are shown for four distinct days, with each day representing the average values of a specific season. Since the real load and PV generation data from Griffith University were used in this study, it is important to note that energy demands during winter and spring were higher than in summer and fall due to the university’s academic calendar. A positive sign indicates BESS charging, while a negative sign represents battery discharging. As observed, during winter and spring, the BESS undergoes more charge–discharge cycles to meet the load demands.

As presented in Table 3, the study scenario without considering reliability enhancement is embedded within the first scenario using the $ϵ$-Constraint method. This approach involves creating the pay-off matrix by deactivating one objective, solving for the other, and repeating the process to determine the maximum and minimum values of each objective. However, according to the Australian Energy Market Commission rules [23], the ENS must be maintained at a minimal level and is treated as a hard constraint that cannot be violated. For instance, if the ENS value is permitted to increase up to 2% of the annual load of the HMG, the resulting optimal Pareto front is illustrated in Figure 5.

5.2. Scenario 2: Without Battery Degradation Consideration

In the second scenario, the results of the proposed planning strategy are compared with those of the HMG planning method that did not account for the cyclic and calendar ageing of the BESS [14]. Although the sizing of the ILC and its stress mitigation were not addressed in the literature, they are considered in this scenario for the sake of a fair comparison. This ensures the proposed method is compared with an existing approach, specifically from the perspective of BESS degradation.

By neglecting the cyclic and calendar ageing of the battery, a smaller capacity for both the BESS and ILC is required, as the BESS can theoretically undergo infinite charge and discharge cycles, reducing the need for a larger ILC to exchange power between sub-grids. However, as illustrated in Figure 6 and Figure 7, the degradation of the BESS and ILC is significantly higher compared to that of the proposed strategy.

5.3. Scenario 3: Constant ILC Capacity

In this scenario, the effectiveness of the proposed planning strategy is compared with that of other existing methods in the literature, [1] that did not consider the optimal sizing of the ILC in the optimal planning of the HMG problem. The simulation study indicates that selecting an ILC capacity below 250 kW results in an infeasible optimisation. Therefore, two different ILC capacities are considered for comparison.

Table 4. Multi-objective planning results with fixed ILC capacity $P^{ILC}$ = 250 kW.

${\mathit{OF}}_1(\mathbf{AUD})$	${\mathit{OF}}_2(\mathbf{kW})$	${\mathit{E}}_{\mathit{AC}}^{\mathit{BESS}}(\mathbf{kWh})$	${\mathit{E}}_{\mathit{DC}}^{\mathit{BESS}}(\mathbf{kWh})$	${\mathit{P}}_{\mathit{A}\mathit{C}}^{\mathit{B}\mathit{E}\mathit{S}\mathit{S}}(\mathbf{k}\mathbf{W})$	${\mathit{P}}_{\mathit{AC}}^{\mathit{BESS}}(\mathbf{kW})$	${\mathit{P}}^{\mathit{ILC}}$
245,923.26	0.00	7439.57	6920.27	190.04	148.50	250.00
246,465.56	0.08	7418.45	6996.79	147.49	167.41	250.00
251,894.05	0.17	7914.88	6783.71	216.04	148.32	250.00
246,728.76	0.25	7666.45	6787.41	146.94	136.34	250.00
252,626.39	0.33	7967.34	6757.19	234.33	148.11	250.00
252,905.03	0.41	7833.00	6903.46	233.59	149.44	250.00
253,483.46	0.50	7908.59	6913.56	183.01	148.23	250.00
253,067.24	0.58	7814.27	6937.50	238.69	146.58	250.00
252,267.08	0.66	7883.38	6816.07	236.42	148.90	250.00
251,956.35	0.75	7859.70	6829.00	222.75	148.22	250.00
250,013.70	0.83	7777.59	6846.36	182.19	136.40	250.00

presents the optimisation results for $P^{ILC}$ = 250 kW, while

Table 5. Multi-objective planning results with fixed ILC capacity $P^{ILC}$ = 1000 kW.

${\mathit{OF}}_1(\mathbf{AUD})$	${\mathit{OF}}_2(\mathbf{kW})$	${\mathit{E}}_{\mathit{AC}}^{\mathit{BESS}}(\mathbf{kWh})$	${\mathit{E}}_{\mathit{DC}}^{\mathit{BESS}}(\mathbf{kWh})$	${\mathit{P}}_{\mathit{AC}}^{\mathit{BESS}}(\mathbf{kW})$	${\mathit{P}}_{\mathit{AC}}^{\mathit{BESS}}(\mathbf{kW})$	${\mathit{P}}^{\mathit{ILC}}$
258,900.83	0.00	7114.27	6936.08	259.63	359.76	1000.00
260,337.32	0.08	7131.80	6910.16	415.08	320.52	1000.00
258,034.73	0.17	7035.89	6998.24	258.89	321.44	1000.00
261,161.14	0.25	6994.50	6952.17	437.38	464.99	1000.00
259,591.17	0.33	7127.35	6667.72	408.10	536.26	1000.00
258,636.64	0.41	6964.81	6893.48	408.40	408.47	1000.00
262,062.78	0.50	7200.79	6929.12	294.63	463.96	1000.00
258,173.78	0.58	7044.86	6867.22	407.65	321.39	1000.00
258,616.69	0.66	7086.48	6940.15	268.20	358.64	1000.00
258,631.79	0.75	7175.13	6798.15	405.30	288.03	1000.00
258,636.73	0.83	7001.62	6904.83	407.21	360.19	1000.00

shows the results for $P^{ILC}$ = 1000 kW. These two values for the nominal power of the ILC were selected because the optimal results presented in Table 3 indicate that the optimal capacity of the ILC is approximately 500 kW. Therefore, one smaller (250 kW) and one larger (1000 kW) value were chosen to compare the optimisation results effectively.

The results in Table 3 and Table 4 indicate that opting for a smaller, non-optimal capacity for the ILC necessitates a larger BESS capacity to supply the loads and meet reliability requirements, thereby increasing the planning costs of the HMG. A comparison between the planning results with an overdesigned ILC capacity, Table 5, and the proposed optimal planning demonstrates that, although selecting a larger ILC capacity slightly reduces the required BESS capacity, the excessive sizing of the ILC imposes high costs on the system.

6. Conclusions

This paper presents a planning strategy for BESSs and ILC in HMG systems. The proposed approach models both the sizing and degradation of the ILC, which have been overlooked in the literature and incorporates them into the optimal planning framework. Both cyclic and calendar ageing of the BESS are modelled and considered in the proposed planning strategy. To minimise cyclic ageing, the energy throughput concept is adopted as a constraint, while calendar ageing is modelled using an empirical method. The ENS reliability index is also included to ensure compliance with the reliability standards set by the Australian Energy Market Commission. The planning problem is formulated as a MOMILP model and solved using the $ϵ$-Constraint method. The effectiveness of the proposed strategy is compared with two widely used approaches from the literature to assess its effectiveness. The results demonstrate the superior technical and economic performance of the proposed method. The uncertainty in the data and the stochastic modelling of the optimal planning system can be explored in future work to assess the impact of forecast uncertainty and the intermittent nature of renewable energy resources.