Sizing of Hybrid PV/Battery/Wind/Diesel Microgrid System Using an Improved Decomposition Multi-Objective Evolutionary Algorithm Considering Uncertainties and Battery Degradation

Abstract

In this paper, a small-scale PV/Wind/Diesel Hybrid Microgrid System (HMS) for the city of Yanbu, Saudi Arabia is optimally designed, considering the uncertainties of renewable energy resources and battery degradation. The optimization problem is formulated as a multi-objective one with two objective functions: the Loss of Power Supply Probability (LPSP) and the Cost of Electricity (COE). An Improved Decomposition Multi-Objective Evolutionary Algorithm (IMOEAD) is proposed and applied to solve this problem. In this approach, different decomposition schemes are combined effectively to achieve better results than the classical MOEA/D approach. Twelve case studies are investigated based on different scenarios and different numbers of houses (5 and 10 houses). Each time, the suggested approach produced a set of solutions that formed a Pareto front (PF). Considering a variety of parameters, the optimal compromise option can be selected by the designer from the PF.

1. Introduction

The ongoing depletion of fossil fuel reserves, increasing environmental concerns, and advancements in power electronic technologies have led to a significant rise in the utilization of Renewable Energy Sources (RESs) globally. Many countries have started using RESs on a large scale and are planning to proceed further in cultivating green energy resources. However, like any other new technology, the production of electrical energy and usage of any RES are challenged by various factors. The precarious behavior of RESs, high expense in the starting phase, and requirement of requisite storage facilities have made it difficult to explore the RESs in islanded areas and rural places. Therefore, it is more practical and suitable to combine the available infrastructure of energy generations with RES technologies. Thus, the Hybrid Microgrid System (HMS) concept has become popular where conventional methods like diesel generators relate to solar-cell- or wind-turbine-generated electrical energy. However, the design, size, capacity, and hybridization policy of HMSs have always been interesting topics for researchers, as it may open the door for high efficiency, reliability, and cost-effectiveness in this arena [1].

Due to their complementary roles in producing power, solar and wind energies are the most harnessed RESs. They are widely accessible worldwide and have substantial environmental advantages. Further development of these two RESs has been boosted and stimulated in recent years by factors such as easy installation, cheap maintenance costs, environmental issues, backup of the traditional energy resources, and convenient laws on RESs. Despite their high capital costs, solar and wind energy facilities have been widely installed worldwide due to their ability to mitigate load demand gaps without carbon emissions gaps [2,3].

A substantial amount of study [4,5,6] has been conducted on the possibility of harnessing RESs with the burning of fossil fuels in order to effectively use RESs. Despite being the world’s largest provider of fossil fuels, the Kingdom of Saudi Arabia still considers wind and solar energy as the two most viable options to traditional sources. Around the Kingdom, particularly in the coastal regions, several feasibility studies have been conducted. The small-scale microgrid model, known as the nanogrid, has been investigated in [7,8] for Hafr Al Batin, a city in the Eastern region of KSA. Furthermore, due to its abundance, solar energy is the region’s go-to substitute for fossil fuels. In addition to the fossil fuel reserve, other nations have also regarded solar and wind energy as the standard sustainable energy sources. For instance, Egypt’s ambitious goal is to use 20% more RESs by 2027 [9], primarily from solar and wind energy. For the cement mill Al-Tafilah in Jordan, the authors of the article designed a hybrid PV–wind system [10]. Another study uses various optimization algorithms to examine the optimal size of freestanding hybrid microgrid systems for a village in the Western Desert of Egypt [11]. A feasibility study of various hybrid energy systems was conducted on five climate regions of Bangladesh [1].

Studies on the application of RESs in Algeria and Brunei, respectively, were described in the works [12,13]. Both articles explored different aspects of RESs as feasible alternatives for future energy needs. According to another study [14], Cambridge Bay, Canada, is economically viable, with an annual capital cost recovery of 11.1%. In contrast, Toamasina, Madagascar is suitable from a sustainable energy point of view, with 76% of the load satisfied and 76% of the energy utilized. Studies conducted on the feasibility of sustainable energy sources in Australia have found that HMSs are a cost-effective solution for moderate loads, including tourist attractions and hotels with grid connections [15,16]. In a similar vein, studies on a remote island in Thailand, such as [17,18], have investigated the effectiveness of hybrid PV/diesel systems [19]. Collectively, these studies and others of a similar nature indicate that integrating renewable energy sources (RESs) with conventional diesel generators could lower energy costs and enhance power system reliability.

The generated power from the RESs is highly variable because of the site dependence and discontinuity of wind and solar energies [20]. Isolated solar or wind energy systems may appear unstable and uneconomical due to their intermittent energy output, as they are unable to consistently meet energy demands [21]. In order to smooth out the unwanted variations, solar and wind energy systems are combined [22]. The consequences of intermittent RES supply can be lessened by incorporating diesel generator and storage systems [23,24]. The storage system is essential in this setup for balancing the intermittency of renewable energy generation and meeting power demand. Such hybrid units’ engineering and dimensioning present extremely difficult technical and financial problems [19]. Under sizing the system will result in power supply shortages when trying to fulfil the specified demand, while an extensive system will cost more money [25]. Therefore, the best HMS unit sizing is crucial for power system design and operation in light of the aforementioned concerns.

The optimal sizing of hybrid microgrid systems has been the subject of a substantial number of studies that have been documented in the literature. Various software programs have been tried for this purpose over time as well. The creation of efficient computational methods is necessary due to a number of drawbacks that each of these instruments suffers from [26]. For this optimization goal, many deterministic techniques including graphical, analytical, numerical, and iterative formulations have been tried [25]. However, numerous hybrid microgrids have also effectively used metaheuristic or nature-inspired heuristic algorithms, leading to enhanced technical and financial efficiency [27].

Despite the disadvantage of coding complexity in hybrid systems, the genetic algorithm (GA) has been used successfully to handle hybrid systems with many parameters [28]. Particle swarm optimization (PSO), which has significantly quicker response times and superior convergence, outperforms GA in other studies. [29]. The optimization problem of reducing COE and LPSP for an HMS was addressed using a multi-objective PSO in [30]. The self-adaptive differential evolution (SDE) technique is suggested to tackle the multi-objective optimization problem in one of the author’s earlier works [31]. Article [32] considered load uncertainty in the problem formulation and solved the issue with the decomposition-based multi-objective algorithm. Other intelligent optimization techniques have been reported in the literature, including the teaching learning-based optimization (TLBO) [33], tabu search (TS) [34], simulated annealing (SA) [35], cuckoo search algorithm (CSA) [36], artificial immune system (AIS) [37], and Grasshopper optimization system (GOA) [25]. Recent review publications [38,39] have presented in-depth evaluations of the optimization tools tested in these studies [40]. Recent review publications [38,39] have presented comprehensive comparative investigations of these tested optimization tools [40]. Another team of academics has reported encouraging advancements in the application of a multi-objective evolutionary algorithm for designing and sizing [41]. Their recent work [42] has investigated different configurations for the design of hybrid sustainable energy systems, introducing a scenario-dominance-based MOEA. In [43], the optimal design of a PV/biomass-based hybrid energy system for a remote hilly area using the discrete grey wolf optimization (GWO) algorithm was investigated.

The HMS has been optimized using the methods indicated above with favorable results. Nevertheless, further developments or alternative algorithms that can address the issue more effectively may enhance existing research outcomes. This paper proposes an enhanced version of the decomposition-based MOEA to address the optimization problem of the HMS. As evidenced in some highlighted works, decomposing MOPs simplifies the problem by generating a set scalar optimization problem. The solution of such a MOP can be achieved using MOEA/D [44] with a suitable decomposition technique. Typical decomposition techniques include weighted sum (WS), weighted Tchebycheff (TCH), and Penalty-based Boundary Intersection (PBI) approaches [45]. Over the years, various variants of these decomposition methods have been developed to address various challenges. For example, some well-known variants of the TCH approach [44] include the augmented achieving scalarizing function (AASF) [46], the weighted-metrics-based methods (WMMs) [47,48], the multiplicative scalarizing function (MSF) factor, and the penalty-based scalarizing function (PSF) [49]. Similarly, notable variants of the PBI approach include the adaptive penalty scheme (APS) [50], the subproblem-based penalty scheme (SPS), the inverted PBI [51,52], and the augmented PBI [53]. While each of these variants offers specific benefits and advantages, hybridizing aspects of them could result in a more robust approach.

In this research, battery degradation concerns and load uncertainty are considered when designing the PV/Wind/Diesel HMS for Yanbu city. Some key features of this article are as follows:

• The paper uses the multi-objective approach of problem formulation because it has the unique Pareto Front (PF) property of presenting the set of solutions of a multi-objective problem in a single shot. Using their expertise, the design requirements, and constraints, the design engineers can choose from any available solution.
• To address the issue of developing HMSs, a novel multi-objective strategy based on an enhanced Decomposition-Based Multi-Objective Evolutionary Algorithm (MOEA/D) is developed in this study. The results obtained from several decomposition methods are combined into a single set of solutions using the proposed novel approach.
• To assess its impact on the completed design, this article models and examines two crucial features: load variability (uncertainty) and battery deterioration.

The robustness of the suggested approach is intended to be ensured by the two test scenarios with 5 and 10 dwellings, respectively. Moreover, the optimization problem considers the COE and LPSP, which are widely used key performance indicators in microgrid planning and operation. The results obtained can provide valuable insights into the optimal allocation of energy resources within the specified limitations, which can aid in the design of microgrids.

The subsequent sections of this paper are organized as follows: Section 2 provides a detailed discussion of the HMS. Section 3 and Section 4 outline the optimization procedure and the suggested strategy of the solution, respectively. Section 5 is designated to presenting the results and their relevant analysis. Finally, Section 6 draws the conclusion.

2. Hybrid Microgrid System Configuration

The configuration of the hybrid microgrid system (HMS) considered in this study is presented in Figure 1 that consists of five major components. In this configuration, the PV plant, wind plant, and the battery bank are the three DC power components. On the contrary, the AC power components include the diesel generator and the domestic load. For the DC-to-AC conversion, an IGBT-based high-power inverter is considered in this HMS model. This inverter along with the AC and DC buses interconnects the generating elements and the load. The optimization problem development will ignore the inverter’s dynamics since it is associated with high frequencies. To monitor and control the power sharing among the elements of the HMS, an energy management system (EMS) is associated with this model. All the elements have a duplex communication link with the EMS, so the EMS continuously receives the information of the state of each component. In return, the EMS sends control signals to each device to ensure optimal performance of the HMS.

2.1. Mathematical Model of the HMS

2.1.1. PV Plant

As characterized by [54], the power output from the PV panel, $P_{PV-out}$, is given by the following equation:

$$P_{PV-out}=P_{N-PV}\times \frac{G}{G_{ref}}\left[1+K_t\left((T_{amb}+\left(0.0256\times G\right))-T_{ref}\right)\right]$$

(1)

where $T_{amb}$ is the ambient temperature, which is 25 °C at Standard Test Conditions (STCs) in this study. $T_{ref}$ represents the surface temperature of the PV cell. The solar irradiance in W/m² is denoted by G. G_ref and K_t are two PV cell constants with values of 1 kW/m² and −3.7 × 10⁻³ °C⁻¹, respectively. $P_{N-PV}$ denotes the rated output power of the PV panel at STCs.

2.1.2. Wind Plant

For the wind plant, a wind turbine is utilized to harness the wind energy to generate electrical power. The output power obtained from the wind turbine relies on the wind speed of a particular site. The turbine height is a crucial factor as the wind speed changes with the variation in turbine height. Using the power-law equation [55], the wind speed can be evaluated as a function of the height of the wind turbine as given below:

$$\frac{v_2}{v_1}={\left(\frac{h_2}{h_1}\right)}^{\alpha }$$

(2)

where v₁ indicates the wind speed at the reference hub height, h₁, and v₂ represents the wind speed corresponding to the desired hub height, h₂. The coefficient of friction is denoted by the power-law exponent, $\alpha$, that depends on different topographical properties of the terrain such as time of the year, height, speed, roughness, and temperature [6,56,57]. According to different research studies, terrains can be classified into several categories [58,59,60]. However, as per the recommendation of IEC standards [61,62], the value of $\alpha$ is 0.11 for extreme wind conditions, whereas it is 0.20 under normal wind conditions. At a particular instant, the wind turbine output power, P_Wind, can be expressed as follows:

$$P_{wind}=\left\{\begin{array}{ll}0& V<V_{cut-in},V>V_{cut-out}\\ V^3\left(\frac{P_{rated}}{V_{rated}^3{-V}_{cut-in}^3}\right)-P_{rated}\left(\frac{V_{cut-in}^3}{V_{rated}^3{-V}_{cut-in}^3}\right)& V_{cut-in}\le V<V_{rated}\\ P_{rated}& V_{rated}\le V<V_{cut-out}\end{array}\right.$$

(3)

where V signifies wind speed at the current time instant and P_rated indicates the rated power. Furthermore, cut-out, cut-in, and rated wind speed are represented by $V_{cut-out}$, $V_{cut-in}$, and $V_{rated}$, respectively.

2.1.3. Battery

Energy storage is a key sector in designing microgrids [63,64]. The battery unit is the third element connected to the DC bus. As shown in the following expression, the battery is typically characterized with the help of its capacity, $C_B$ [65]:

$$C_B=\frac{{AD E}_L}{DOD {\eta }_{inv}{\eta }_b}$$

(4)

where $E_L$ denotes the load and AD represents the autonomy days. In Equation (4), DOD indicates the depth of discharge, which is set to 80%. The inverter efficiency, η_inv, is set to 92%, whereas the battery efficiency, η_b, is set to 85%.

2.1.4. Diesel Generator

The standalone diesel generator connected to the AC bus works as a backup or secondary source to enhance the operational stability of the HMS. It plays a crucial role in satisfying the load demand, especially when the RESs are not capable of doing so. As discussed in [66], the generator efficiency remains low when the output power is low. Hence, the diesel generator needs to be operated around the nominal output power for better energy utilization. In case of inconsistent load demand, this strategy provides secured operation against the power fluctuation [67]. The fuel consumption of the generator, q(t), is expressed by the following equation [68,69]:

$$q\left(t\right)=aP\left(t\right)+b{P}_{rated}$$

(5)

where a and b are the coefficients of fuel consumption and their respective values are considered to be 0.246 and 0.08415 [70]. Furthermore, the output power of the diesel generator for a given time instant is denoted by P(t), whereas the rated power is represented by P_rated.

Diesel generator net efficiency, ${\eta }_T$, can be expressed using the following relation [71]:

$${\eta }_T={\eta }_B\times {\eta }_G$$

(6)

where ${\eta }_G$ is efficiency of the generator and ${\eta }_B$ represents the thermal brake efficiency.

2.1.5. Inverter

For the HMS configuration considered in this study, the inverter is a vital operational component. It is mainly responsible for converting DC power to AC to satisfy necessary power requirements of the AC loads. The efficiency of the inverter, ${\eta }_{inv}$, is approximated through the following expression [72,73]:

$${\eta }_{inv}=\frac{P}{P+P_0+k{P}^2}$$

(7)

where

$$P=\frac{P_{out}}{P_{in}}$$

(8)

$$P_0=1-99{\left(\frac{10}{{\eta }_{10}}-\frac{1}{{\eta }_{100}}-9\right)}^2$$

(9)

$$k=\frac{1}{{\eta }_{100}}-P_0$$

(10)

where ${\eta }_{10}$ is a constant representing the inverter efficiency at 10% of its nominal power and ${\eta }_{100}$ is the corresponding value at 100%. These quantities are typically provided by the manufacturers.

2.1.6. Management Strategy of EMS

If all the components of the HMS are in close vicinity of the power consumption, then it can be considered that minimal electrical losses will be associated with the distribution of power. The goal of the EMS in the proposed HMS is to implement a management strategy to provide optimal power sharing between all the elements of the HMS. The management strategy to be implemented by the EMS is highlighted in the flowchart shown in Figure 2 [30]. According to this strategy, power generated by the RES (PV plant, wind plant) will be utilized on a priority basis to satisfy the load demand. After meeting the load demands, any excess energy available from these renewable plants will be used in charging the battery units. In case the load demand is higher than the combined power produced by the RES, the EMS will try to fulfil the remaining demand through the battery bank. In an extreme scenario, when all the elements connected with the DC bus fail to meet the load requirements, the diesel generator will be operated to overcome the shortage of power. If any surplus energy is available from the diesel generator, it will be utilized in recharging the battery bank for future requirements. In Figure 2, $E_b\left(t\right)$ and $E_b\left(t-1\right)$ denote the energy of the battery at the current time instant and the previous time instant, respectively. $E_{ch}\left(t\right)$ stands for charging energy of the battery and $E_{dch}\left(t\right)$ denotes its discharging energy The excess energy that is dumped at the current time instant is represented by $E_{dump}\left(t\right)$. $E_{bmin}$ and $E_{bmax}$, respectively, denote the minimum and maximum storage capacities of the battery. The equations in the preceding sections contain the rest of the symbols and their corresponding explanation.

2.2. Uncertainty

Dealing with different kinds of uncertainty proves to be a challenge in various fields of study such as practical engineering, general sciences, and economics. Uncertainty results from information that is not known and difficult to predict beforehand. It is almost impossible to figure out such information with complete certainty owing to limitations resulting from inaccuracies in sensor data to the limitations of physical laws of the universe. To be specific, there is no guarantee in the prediction of uncertain events. There must be a margin between the timing of an event and its likelihood of occurrence.

In practice, multiple approaches are adopted to represent uncertainty. Setting a tolerance range, defined as the percentage deviation from the obtained or measured value, is one of the commonly used methods.

2.2.1. Uncertainty in Microgrids

Uncertainties are inherent in all the elements of a microgrid system. Thus, in the domain of optimization-based research on microgrids, uncertainties play a crucial role. The formulation of the optimization problem is adversely affected by the associated uncertainties. From load demands to the generation of power, market prices to the consumer reaction to bills, and the accessibility of generating units to the isolation of microgrids, all these are non-deterministic events. When the uncertainty in the electric market prices is considered, the governing authority experiences a jeopardizing situation [19]. Thus, deterministic modeling of such uncertainties is mathematically intractable. Therefore, the pre-eminence of uncertainties in microgrids makes it necessary to incorporate it in the system model by adopting stochastic methods [74].

2.2.2. Stochastic and Deterministic Models

A predictable model can be seen as a precursor to a probabilistic one. In the former, the required parameters are already known, or they can be determined with ease, and the output can be accurately realized from the given set of inputs. Results obtained from such models may deviate from the realistic scenario where unpredictable changes might occur to different parameters. However, creating a predictable model can give a reasonable estimation and establish the groundwork for probabilistic modeling. It is feasible that predictable models may change over time. The model is referred to as a steady-state deterministic process if there is no fluctuation with regard to time. By contrast, a time series of steady-state deterministic processes is what is used when the input parameters of a deterministic model change incrementally over time. A dynamic deterministic model falls under the last group; it is dependent on several incremental series (such time and space) and has known input values for each state. This category is typically useful in forecasting studies [74].

The focus of stochastic modeling is to utilize various techniques to integrate the randomness inherent in the process to be modeled. For a given state, the inputs are not fixed; instead, there is a margin of uncertainty. A parameter might have any value within a specified range. Therefore, it is likely to obtain a different set of results for every independent run of a stochastic model. As a result, a probability distribution can be obtained through this process. A few examples of the stochastic modeling include the Poisson, Brownian, and Markov processes.

The Monte Carlo simulation is a mathematical technique useful in predicting the probability of different outcomes of a process with parameters that are random in nature. To incorporate the randomness, every input parameter randomly takes a value within the specified range of parametric uncertainty. The model is executed for a large number of iterations with a randomly generated unique set of inputs for every run. Finally, a statistical toolset can be used to obtain a probabilistic output. This technique is commonly employed in various applications, including optimization and numerical integration [74].

2.3. Battery Degradation

Battery degradation is a natural process by which the capacity of the battery to store energy reduces over time. For a particular project, it is not desirable and not cost-effective to replace batteries within the project’s lifetime. Thus, incorporating a battery degradation model is important for planning studies to ensure the required battery storage without the replacement of batteries, preventing additional investment costs. The DOD, allowable number of charge/discharge cycles, and the calendar lifetime of battery storage are the key factors to be considered for modeling the battery degradation [75]. A mixed integer programming (MIP)-based approach was adopted in [76], for determining the optimal size and DOD of a standalone microgrid, to minimize expansion costs. In [77], the battery degradation factors were incorporated into the formulation of a probabilistic mixed integer linear programming (MILP) approach for optimal integration of RESs and energy storage for the minimization of energy costs in fast charging stations. An improved MILP-based method was introduced in [78] to determine the optimal size of battery storage along with DOD and replacement year, taking the service life and capacity degradation of battery storage into consideration. In [79], the battery life degradation model was considered for a pragmatic MILP-based approach that can efficiently determine the optimal DOD and the number of charge/discharge cycles over a project’s lifetime.

3. Optimization Process

3.1. Problem Formulation

For optimal operation of the HMS, a multi-objective optimization approach is adopted in this study. A multi-objective optimization problem (MOP) can be typically represented as

$$\begin{array}{ll}\hfill \mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& F\left(\mathbf{x}\right)={\left[f_1\left(\mathbf{x}\right),f_2\left(\mathbf{x}\right),\dots ,f_k\left(\mathbf{x}\right)\right]}^T\hfill \\ \hfill \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o}& g\left(\mathbf{x}\right)=0\hfill \\ \hfill \mathrm{a}\mathrm{n}\mathrm{d}& h\left(\mathbf{x}\right)\le 0\hfill \end{array}$$

(11)

where x is a vector of the design variables associated with the problem. F(x) represents the vector consisting of the individual objective functions, $f_1\left(\mathbf{x}\right),f_2\left(\mathbf{x}\right),\dots ,f_k\left(\mathbf{x}\right)$. Vectors $g\left(\mathbf{x}\right)$ and $h\left(\mathbf{x}\right)$ represent the equality and inequality constraint sets, respectively.

3.2. Objective Functions

To ensure the cost-effectiveness and reliability of power supply of the HMS, two objective functions are considered in this study. They are the COE and the LPSP. Minimizing these two objective functions satisfying the associated constraints of the design variables can ensure the most optimum performance of the HMS.

3.2.1. Cost of Electricity (COE)

The definition of the COE is the average cost of generating usable electrical energy by a power system in its lifespan. In the context of hybrid energy systems, COE is commonly used to evaluate the economic feasibility of different asset configurations. The key factor in COE analysis for hybrid energy systems is the net profit cost (NPC), which includes capital costs, operation and maintenance (O&M) costs, and replacement costs. Although the initial capital cost is high for all renewable energy sources, they offer benefits such as high reliability, low O&M costs, and zero fuel expenses. To simplify the analysis, the procurement costs of PV panels, batteries, wind turbines, diesel generators, and inverters are considered as the cost components of the hybrid energy system in this study.

The COE can be determined using Equations (12) and (13) [29,80], where its unit is USD/kWh:

$$\mathrm{C}\mathrm{O}\mathrm{E}=\frac{Total NPC}{{\sum }_{h=1}^{h=8760}{P}_{load}}\times \mathrm{C}\mathrm{R}\mathrm{F}$$

(12)

where $P_{load}$ denotes the consumed power per hour and CRF stands for the capital recovery factor, which can be determined by

$$\mathrm{C}\mathrm{R}\mathrm{F}=\frac{i{(1+i)}^n}{{(1+i)}^n-1}$$

(13)

where i represents the interest rate per annum for n number of years.

3.2.2. Loss of Power Supply Probability (LPSP)

LPSP is a significant statistical tool in quantifying the reliable supply of power in a system. It denotes the probability of not fulfilling the load demand, due to a deficient power supply. This deficiency of power can result from technical failure, uncertainty in the availability of RESs, and economic constraints. LPSP is generally quantified as the proportion of energy deficit to the total energy requirement over an extended duration as given in the following expression [81,82]:

$$\mathrm{L}\mathrm{P}\mathrm{S}\mathrm{P}=\frac{\sum (P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}-P_{\mathrm{P}\mathrm{V}}-P_{\mathrm{w}}+P_{\mathrm{S}\mathrm{O}\mathrm{C} \mathrm{m}\mathrm{i}\mathrm{n}}+P_{\mathrm{d}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{l}})}{\sum P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}}$$

(14)

It is worth noting that for the analysis of system reliability, it is assumed that the overall load requirement is greater than the total generated power:

$$P_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}\left(t\right)>P_{\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\left(t\right)$$

(15)

3.3. The Constraints for Optimization

In this study, one of the constraints is the renewable factor (RF). The RF is the measure to discriminate between the amount of power produced by renewable and non-RESs. Thus, the contribution of the RES in the total power supply can be estimated by this factor. RF can be expressed as follows:

$$\mathrm{R}\mathrm{F}\left(\mathrm{\%}\right)=\left(1-\frac{\sum P_{\mathrm{d}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{l}}}{\sum P_{\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{s}}}\right)\times 100$$

(16)

An RF value of 100%, which shows that the renewable energy resources satisfy the total load demand, is desirable.

3.4. Design Variables

The design variables considered for this study are the number of autonomy days (N_AD), wind turbines (N_WT), and diesel generators (N_Diesel). In addition, the amount of nominal power from the PV panels is also taken as a variable.

3.5. Formulating the Uncertainty

As mentioned earlier, the design of microgrid systems includes the necessity to encounter some uncertain parameters such as the load. This load uncertainty can be formulated with the help of the following expression [74]:

$$P_{load}\left(t\right)=P_{load}\left(t\right)+x\mathrm{\%}·P_{load}\left(t\right)·{\underset{\_}{u}}^d\left(t\right)-x\mathrm{\%}·P_{load}\left(t\right)·{\overline{u}}^d\left(t\right)$$

(17)

$${\underset{\_}{u}}^d\left(t\right)+{\overline{u}}^d\left(t\right)\le 1$$

(18)

where ${\underset{\_}{u}}^d\left(t\right)$ and ${\overline{u}}^d\left(t\right)$ are binary values.

Therefore, randomly changing $x\mathrm{\%}$ of the real value of the load variable provides a simple way of simulating the uncertainty. Basically, the uncertainty is introduced in each time instant either by increasing or decreasing the load by $x\mathrm{\%}$.

3.6. Degradation Formulation

The charge–discharge cycles and DOD per cycle of batteries are the major elements contributing to their degradation [79]. The degradation is considered in this study to impact the capacity, $C_B$, of the batteries as follows:

$$C_B=\frac{{AD\mathrm{*}E}_L}{DOD {\eta }_{inv}{\eta }_b}\times \mathrm{D}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{\%}$$

(19)

where degradation is denoted in percentage over the batteries’ lifetime and it is selected to be 75% in this study.

3.7. Problem Solution Using Decomposition-Based Multi-Objective Evolutionary Algorithm (MOEA/D)

MOEA/D is utilized to break down the MOP specified in (11) into several objective functions for resolution [44]. This decomposition, achieved using any appropriate decomposition technique, leads to several competing objectives. Typically, the selected decomposition method generates scalar optimization problems that are solved using Pareto optimal vectors. The most commonly used decomposition methods are the weighted sum (WS), weighted Tchebycheff (TCH), and Penalty-based Boundary Intersection (PBI) approaches [45]. In this work, we briefly discuss the WS approach but focus more on and adopt the PBI approach, the TCH approach, and their extensions, as discussed in the following subsections.

3.7.1. The Weighted Sum Approach

The weighted sum (WS) is the convex combination of the different objectives. If we define the weight vector as $\mathit{\lambda }={\left({\lambda }_1,{\lambda }_2,\dots {\lambda }_m\right)}^T$, where ${\lambda }_i\ge 0 \forall i\in \{1,\dots ,m\}$ and ${\sum }_{i=1}^m{\lambda }_i=1$, a weighted sum is defined as

$$\begin{array}{cc}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& {\mathrm{g}}^{\mathrm{w}\mathrm{s}}\left(\mathit{x}|\mathit{\lambda },{\mathit{z}}^{\mathbf{*}}\right)={\displaystyle \sum _{i=1}^m}{\lambda }_i{f}_i\left(\mathit{x}\right)\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o}& \mathit{x}\in \mathsf{\Omega }\end{array}$$

(20)

3.7.2. The Weighted Tchebycheff (TCH) Approach

The Tchebycheff approach is robust to the shape of Pareto fronts and can be presented as the following scalar optimization problem:

$$\begin{array}{c}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e} g^{tch}\left(\mathit{x}|\mathit{\lambda },{\mathit{z}}^{\mathrm{*}}\right)=\underset{1\le i\le m}{\max }\left\{{\lambda }_i\left|f_i\left(x\right)-z_i^{\mathrm{*}}\right|\right\}\hfill \\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o} \mathit{x}\in \mathsf{\Omega }\hfill \end{array}$$

(21)

where $z^{\ast }$ is the reference point combining the smallest values of all the objectives $(z_1^{\ast },\dots ,z_m^{\ast })$: $z_i^{\ast }=\mathrm{m}\mathrm{i}\mathrm{n}\left\{f_i\left(\mathit{x}\right)|\mathit{x}\in \mathsf{\Omega }\right\}$ for $i=1,\dots m.$

3.7.3. The Normalized TCH Version (NTCH)

The TCH approach suffers from the non-smoothness of the aggregation function for continuous MOPs. Moreover, for MOPs with three or more objectives, there is a noticeable degradation in uniformity of the aggregated solutions obtained using TCH compared to PBI, which is exasperated when the number of weight vectors is insufficient [44]. Hence, objective normalization is often employed to improve the solutions in non-uniformly scaled objectives. A typical normalization technique of the objectives $f^i:i=1,\dots ,m$ with their normalized versions ${\overline{f}}^i\in \left[0,1\right]$ is

$${\overline{f}}^i=\frac{f_i-z_i^{\mathrm{*}}}{z_i^{nad}-z_i^{\mathrm{*}}}$$

(22)

where $z^{nad}={\left(z_1^{nad},\dots ,z_m^{nad}\right)}^T$ is the nadir point in the objective space. $z^{nad}$ sets the maximum limit of the PF, i.e., $z_i^{nad}=\max \left\{f_i\left(\mathit{x}\right)|\mathit{x}\in PS\right\}$.

3.7.4. The Modified TCH Version (MTCH)

The MTCH approach is formulated by the following formula:

$$g^{te}\left(\mathit{x}|{\lambda }^j,z^{\mathrm{*}}\right)=\underset{1\le i\le m}{\min }\left\{\frac{\left|f_i\left(\mathit{x}\right)-z_i^{\mathrm{*}}\right|}{{\lambda }_i^j}\right\}$$

(23)

3.7.5. Variants of the TCH Approach

For the TCH-based methods, the shape of the contour line primarily depends on the distance measured with reference to the ideal point. The Tchebycheff distance used in the original TCH method leads to contour lines like the Pareto dominance relation for identifying better solutions [46] and cannot distinguish weakly dominated solutions. To address the issues associated with the TCH approach for subproblem formulation based on the shape of the contour lines and to improve the regions that define solutions, various modifications to the traditional TCH have been proposed over the years. Each of these variants, discussed next, tweaks the contour lines to achieve some improvements by adapting the weighted metric parameter. The contour lines for the different variants of TCH are shown in Figure 3.

Augmented Achievement Scalarizing Function (AASF)

The augmented achieving scalarizing function (AASF) [46] was introduced to tweak the opening angle of the contour lines, as shown in Figure 3b. The function is given as

$$\underset{x\in \mathsf{\Omega }}{\min }{g}^{aasf}\left(\mathit{x}|\mathit{\lambda },{\mathit{z}}^{\mathbf{*}}\right)=\underset{1\le i\le m}{\max }\left\{\frac{\left|f_i\left(x\right)-z_i^{\mathrm{*}}\right|}{{\lambda }_i^j}\right\}+\rho \left\{\frac{\left|f_i\left(x\right)-z_i^{\mathrm{*}}\right|}{{\lambda }_i^j}\right\}$$

(24)

The parameter $\rho$ is used to tweak the contour line’s opening angle. A major drawback of this approach is the lack of thumb rules for the choice of $\rho$ to control the improvement region. Applications of this are found in [46,83].

Weighted-Metrics-Based (WMM) Methods

The crispy improvement region of TCH leads to a loss of population diversity. Hence, weighted metrics methods have been employed to adapt the contour lines’ shape, depicted in Figure 3c, to the subproblems’ respective local regions. A typical function that achieves this is the weighted metrics method [47,48]:

$$\underset{x\in \mathsf{\Omega }}{\min }{g}^{wmm}\left(\mathit{x}|\mathit{\lambda },{\mathit{z}}^{\mathbf{*}}\right)={\left({\displaystyle \sum _{i=1}^m}{\lambda }_i{\left|f_i\left(x\right)-z_i^{\mathrm{*}}\right|}^{\theta }\right)}^{\frac{1}{\theta }}$$

(25)

The parameter $\theta$ controls the geometrical characteristics of the contour lines. The improvement region for each subproblem can be refined via the adaptive tuning of $\theta$ from a predefined set $\mathsf{\Theta }=\{1,\mathrm{2,3},\dots ,10,\infty \}$. This application can be found in MOEA/D-Par [48] and MOEA/D-D-Pas [47].

Multiplicative Scalarizing Function (MSF)

The multiplicative scalarizing function (MSF) factor proposed by [49] is similar to the WMM. Let $\alpha$ be the parameter that controls the shape of the MSF contour, and then the MSF is defined as [49]

$$\underset{x\in \mathsf{\Omega }}{\min }{g}^{msf}\left(\mathit{x}|\mathit{\lambda },{\mathit{z}}^{\mathbf{*}}\right)=\frac{{\left[{\mathrm{m}\mathrm{a}\mathrm{x}}_{1\le i\le m}\left(\frac{1}{{\lambda }_i}\left|f_i\left(x\right)-z_i^{\mathrm{*}}\right|\right)\right]}^{1+\alpha }}{{\left[{\mathrm{m}\mathrm{i}\mathrm{n}}_{1\le i\le m}\left(\frac{1}{{\lambda }_i}\left|f_i\left(x\right)-z_i^{\mathrm{*}}\right|\right)\right]}^{\alpha }}$$

(26)

This becomes TCH when $\alpha =0$ but overlaps with the corresponding weight vector when $\alpha =+\infty$. The parameter $\alpha$ can be made adaptive using [79]:

$${\alpha }^i=\beta (1-g/G_{max})\left\{m\times \underset{1<\mathrm{j}<\mathrm{m}}{\min }\left({\lambda }_j^i\right)\right\}$$

(27)

The parameter ${\alpha }^i\in \alpha$ is the value associated with the $i$-th subproblem.

Penalty-Based Scalarizing Function (PSF)

The penalty-based scalarizing function (PSF) was also proposed by [49] as described by the following equation:

$$\underset{x\in \mathsf{\Omega }}{\min }{g}^{psf}\left(\mathit{x}|\mathit{\lambda },{\mathit{z}}^{\mathbf{*}}\right)={\mathrm{m}\mathrm{a}\mathrm{x}}_{1\le i\le m}\left(\frac{1}{{\lambda }_i}\left|f_i\left(x\right)-z_i^{\mathrm{*}}\right|\right)+\alpha d$$

(28)

where $\alpha$ is a parameter that balances the trade-off between convergence and diversity, and $d=‖\mathit{F}\left(\mathit{x}\right)-{\mathit{z}}^{\ast }+d_1\mathit{\lambda }‖:d_1=‖{\left(\mathit{F}\left(\mathit{x}\right)-{\mathit{z}}^{\mathit{*}}\right)}^T\mathit{\lambda }‖/‖\mathit{\lambda }‖$, which is defined similarly to $d_2$ in the PBI. The contour lines for PSF are presented in [49]. The contour lines for PSF are shown in Figure 3d.

3.7.6. The Penalty-Based Boundary Intersection (PBI) Approach

The PBI technique is a type of boundary intersection (BI) method used for decomposing MOPs. These methods were initially developed for continuous MOPs and are effective with non-concave PFs. Equations (28) and (29) define the BI and PBI approaches, respectively, and their corresponding geometric interpretations are shown in Figure 4 [32]. In the BI method, the objective is to find the points of intersection on the leftmost boundary set of lines, which, if evenly distributed, can sufficiently approximate the entire PF.

$$\begin{array}{ll}\hfill \mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& {\mathrm{g}}^{\mathrm{b}\mathrm{i}}\left(\mathit{x}|\mathit{\lambda },{\mathit{z}}^{\mathbf{*}}\right)={\displaystyle \sum _{i=1}^m}{\lambda }_i{f}_i\left(\mathit{x}\right)\\ \hfill \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o}& F\left(\mathit{x}\right)-{\mathit{z}}^{\mathrm{*}}=d\mathit{\lambda },\\ \hfill \mathrm{a}\mathrm{n}\mathrm{d}& \mathit{x}\in \mathsf{\Omega }\end{array}$$

(29)

where $\mathit{w}$ is the weight vector and ${\mathit{z}}^{\ast }$ is the reference point. $F\left(x\right)$ is constrained to remain on the line while $L$ is constrained to remain along the orientation of $\mathit{\lambda }$ by the equation $F\left(\mathit{x}\right)-{\mathit{z}}^{\ast }=d\mathit{\lambda }$, which also ensures that it passes ${\mathit{z}}^{\ast }$. However, the BI approach is unable to deal with equality constraints.

The PBI method uses a penalty technique to handle equality constraints and is expressed in the following equation.

$$\begin{array}{cc}\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}& {\mathrm{g}}^{\mathrm{p}\mathrm{b}\mathrm{i}}\left(\mathit{x}|,{\mathit{z}}^{\mathrm{*}}\right)=d_1+\theta d_2\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o}& \mathit{x}\in \mathsf{\Omega }\end{array}$$

(30)

where $d_1=\frac{‖{\left(F\left(\mathit{x}\right)-{\mathit{z}}^{\ast }\right)}^T\mathit{\lambda }‖}{‖\mathit{\lambda }‖}$ is the distance between $z^{\ast }$ and $y-$ (the projection of $F\left(\mathit{x}\right)$ on the line $L$); $d_2=‖F\left(\mathit{x}\right)-(z^{\ast }+d_1\mathit{\lambda })‖$ is the distance between $F\left(x\right)$ and $L$; and the penalty parameter, $\theta >0$, can be defined by the user. Choosing an appropriate value of $\theta$ results in outcomes that are similar to those obtained with the BI method. When employed in MOP, the PBI approach results in the equitable allocation and convergence of optimal solutions.

3.7.7. Variants of the PBI Approach

These variants focus on the contour line shape and the improvement region’s size that defines better solutions. The smaller improvement region emphasizes diversity while the converse emphasizes convergence. For PBI, the improvement region is controlled by the parameter $\theta$ from Equation (28). As observed from the equation, it functions like the TCH approach or WS approach when $\theta =1.0$ or $\theta =0.0$, respectively. The significance of the improvement increases with decreasing values of $\theta .$ Hence, an approach for developing PBI variants is by the adaptation of $\theta$ with respect to the shape of the Pareto Front [50,84]. The contours for some variants of PBI are shown in Figure 5.

Adaptive Penalty Scheme (APS)

In the adaptive penalty scheme (APS) [50], the penalty factor, $\theta ,$ increases cautiously as the evolution progresses according to the equation

$$\theta ={\theta }_{min}+\left({\theta }_{max}-{\theta }_{min}\right)\frac{t}{t_{max}}$$

(31)

where $t$ represents the generation count with maximum generation count $t_{max}$: $\left[{\theta }_{min}{\theta }_{max}\right]=\left[1.0 10.0\right]$. APS strives to facilitate convergence at an earlier stage of the evolution and then gradually focuses on diversity later in the evolution process. Schemes based on APS include MOEA/D-PaP [84], NSGA-III-AASF, and NSGA-III-EPBI [85].

Subproblem-Based Penalty Scheme (SPS)

Another method of adapting the penalty is the subproblem-based penalty scheme (SPS). As the name implies, $\theta$ is independently assigned to each of the subproblems such that for the $i$th subproblem:

$${\theta }_i=e^{a{b}_i},b_i=\underset{1\le j\le m}{\max }\left\{{\lambda }_i^j\right\}-\underset{1\le j\le m}{\min }\left\{{\lambda }_i^j\right\}$$

(32)

where the parameters ${\theta }_i$ and ${\lambda }_i^j$ are the penalty for the $i$-th subproblem and $j$-th element of the $i$-th weight, respectively. The variable $a$ is the scaling factor that controls the size of the penalty.

Inverted PBI (IPBI)

The inverted PBI adopts an approach that does not adapt the penalty term. Instead, it uses the equation [51,52]:

$$\underset{x\in \mathsf{\Omega }}{\min }{g}^{ipbi}\left(\mathit{x}|\mathit{\lambda },{\mathit{z}}^{\mathit{n}\mathit{a}\mathit{d}}\right)=d_1^{nad}-\theta d_2^{nad}$$

(33)

where $d_1^{nad}=‖{\mathit{z}}^{nad}-\mathit{F}\left(\mathit{x}\right)\mathit{\lambda }‖/‖\mathit{w}\mathit{\lambda }‖$ and $d_2^{nad}=‖{\mathit{z}}^{nad}-\left(\mathit{F}\left(\mathit{x}+{\mathit{d}}_1^{\mathit{n}\mathit{a}\mathit{d}}\mathit{\lambda }/‖\mathit{\lambda }‖\right)\right)‖$. IPBI functions by pushing away the solution as far as possible from ${\mathit{z}}^{nad}$. As such, it often has similar characteristics to the traditional PBI but with the ability to search over a more comprehensive objective space. The contour lines for the IPBI are shown in Figure 5b.

The Augmented PBI (APBI)

The augmented format of the PBI uses augmented multiple distance metrics as follows:

$$\underset{x\in \mathsf{\Omega }}{\min }y\left(\mathit{x}|{\mathit{n}}^{\mathbf{*}},{\mathit{z}}^{\mathbf{*}}\right)=h(\overline{\mathit{F}}\left(\mathit{x}|{\mathit{n}}^{\mathrm{*}},{\mathit{z}}^{\mathrm{*}}\right)=d_1+{\theta }_1{d}_2^2+{\theta }_2{d}_2^4$$

(34)

where the parameters ${\theta }_1>0$ and ${\theta }_2>0$ control the underlying subproblem’s search attitude via the distribution and shape of the curvature contours and opening angle. In practice, a Gaussian process regression model is used to set ${\theta }_1$ and ${\theta }_2$ based on the manifold structure of the underlying PF. An example of this variant is the MOEA/D-LTD [53]. The contour lines for LTD with fixed ${\theta }_2$ and ${\theta }_1$ are shown in Figure 5c,d, respectively.

3.7.8. The General Framework for MOEA/D

MOEA/D utilizes an appropriate decomposition method to decompose the MOP based on the given problem. In this scenario, it is assumed that the TCH approach is employed; however, any other appropriate decomposition technique can be readily substituted.

The $j$th objective function in the TCH decomposition method is defined as

$$g^{te}\left(\mathit{x}|{\mathit{\lambda }}^{\mathit{j}},{\mathit{z}}^{\mathbf{*}}\right)=\underset{1\le i\le m}{\min }\left\{{\lambda }_i^j\left|f_i\left(x\right)-z_i^{\mathrm{*}}\right|\right\}$$

(35)

where ${\mathit{\lambda }}^j=({\lambda }_1^j,{\lambda }_2^j,\dots ,{\lambda }_m^j)$ represents the $jth$ weight vector from a set of evenly spread weight vectors ${\lambda }^1,\dots ,{\lambda }^N$, and ${\mathit{z}}^{\mathit{*}}$ represents the reference point. The neighborhood weight vector, ${\lambda }^i$, is defined by the set of nearest weight vectors, $\left\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N\right\}.$ Usually, only the most recent solutions of neighboring subproblems are used for optimization. At each iteration, in the TCH-decomposed MOEA/D, the maintenance of certain data is described in [44]. The maintained information includes:

1.

The population of the most recent solution to the N subproblems, $x^1,\dots ,x^N$.

2.

The F values of these recent solutions, $F{V}^1,\dots F{V}^N:$ $F{V}^i=F\left(x^i\right)$.

3.

The set of optimal values for the objectives discovered so far, $z=(z_1,\dots z_m)$.

4.

An external population, $EP,$ comprising solutions that are nondominated.

5.

Refer to Algorithm 1 for the details of the MOEA/D algorithm [44].

With the help of the Euclidean distance parameter, the distance between two vectors is calculated. If objective normalization is performed, it is not essential to compute $z^{nad}$ and $z^{\ast }$ beforehand. In this study, $z^{\ast }$ is substituted with $z$, while $z^{nad}$ is replaced by ${\tilde{z}}_i^{nad}$, which is the maximum value of objective ($f_i)$ in the current population. Consequently, $g^{te}$ is substituted with a new value, given by (36), during the update of neighboring solutions step.

$$\underset{1\le i\le m}{\min }\left\{{\lambda }_i\left|\frac{f_i-z_i}{{\tilde{z}}_i^{nad}-z_i}\right|\right\}$$

(36)

Algorithm 1: MOEA/D
	Input:  (1) MOP (11);  (2) Stopping criteria $\left(e_{th}\right)$;  (3) N: Number of MOEA/D subproblems $\left(N\right)$;  (4) N uniform spread of weight vectors $\{{\lambda }^1,\dots ,{\lambda }^N\};$  (5) T: no. of weight vectors in the neighborhood of each weight vector Output: $\mathit{E}\mathit{P}$
1	Initialization: $EP=\mathrm{Ø}$
2	Compute the Euclidean distance between any two weight vectors and find the T closest weight
	vectors to each weight vector.
3	for $i\leftarrow 1 \mathrm{t}\mathrm{o} N$ do
4	$B\left(i\right)=\left\{i_1,\dots ,i_T\right\}:{\lambda }^{i_1},\dots ,{\lambda }^{i_T}$ – the T closest weight vectors to ${\lambda }^i$.
5	end for
6	Generate initial population $x^1,\dots ,x^N$ randomly or by the problem-specific method.
7	Set $F{V}^i=F\left(x^i\right)$.
8	Initialize $z={\left(z_1,\dots ,z_m\right)}^T$ by a problem-specific method
9	while the stopping criteria is not met do
10	Update:
11	for do
12	Reproduction:
13	Randomly select two indexes $k,l$ from $B\left(i\right)$, and generate a new solution $y$
	from $x^k$ and $x^l$ by using genetic operators
14	Improvement:
15	Apply a problem-specific repair/improvement heuristic on $y$ to produce $y^{\prime }$
16	Update $z:$
17	for $i\leftarrow 1$to $m$ do
18	if $z_j>f_j\left(y^{\prime }\right)$ then
19	$z_j=f_j\left(y^{\prime }\right)$
20	end if
21	end for
22	Update Neighboring Solutions:
23	for $i\in B\left(i\right)$ do
24	if $g^{te}\left(y^{\prime }|{\lambda }^j,z\right)\le g^{te}\left(x^j\right|{\lambda }^j,z)$
25	set $x^j=y^{\prime }$ and $F{V}^j=F\left(y^{\prime }\right)$
26	end if
27	end for
28	Update EP:
29	Remove from EP all the vectors dominated by $F\left(y^{\prime }\right)$
30	Add $F\left(y^{\prime }\right)$ to EP if no vectors in EP dominate $F\left(y^{\prime }\right)$
31	end for
32	end while

4. The Proposed IMOEAD Approach: Description and Implementation

As aforesaid, in this paper, a new improved MOEA/D is proposed, developed, and implemented to optimally size a hybrid PV/Battery/Wind/Diesel microgrid system. This proposed approach combines the multiple decomposition techniques presented in Section 3.7 and it is noted as IMOEAD. It is implemented to solve the problem at hand, as shown in the flowchart of Figure 6.

It starts by collecting different data of the microgrid and its different components. Subsequently, the internal optimization parameters, such as the population size and maximum iteration count, can be specified by the user (designer). The population is then randomly initialized and generated in the search space. Then, 13 threads are open in parallel; for each one, a different decomposition technique is run separately (from the first to the thirteenth thread, ES, TCH, NTCH, MTCH, AASF, WMM, MSF, PSF, PBI, APS, SPS, IPBI, and ABPI are run in parallel). The population size for each algorithm is the same as the first one given in the initial data. Once the updated population for each algorithm is obtained, they are all combined in one population. The nondominated solutions are determined from this population. And the process runs again and again until the termination criterion is updated. Finally, the optimal results are displayed, and the designer takes the needed further steps to exploit these results.

5. Application Results and Discussion

Yanbu is an industrial city in western Saudi Arabia, which has ample renewable energy resources. Yanbu is located at a latitude of 24°05′20″ North and a longitude of 38°03′49″ in the coastal area of the Red Sea [86,87]. Therefore, it is chosen as the case study region for demonstrating the proposed optimization approach. Based on previous research works, the average annual wind speed of 3.53 m/s and solar irradiation of 5.95 kWh/m²/day [88] are used in the problem formulation. Real weather data for Yanbu are utilized in the optimization process. The temperature in Yanbu ranges from 15 °C to 40 °C, with an average annual temperature of approximately 29 °C. The per-hour wind speed and solar irradiation data are illustrated in Figure 7a,b, respectively [86,87,88].

5.1. Investigated Cases

In this paper, twelve different cases are investigated. The lower and upper range of values for PV, N_WT, N_Diesel, and N_AD are considered as [15, 45], [0, 10], [1, 4], and [1, 5], respectively. The first six cases investigate a microgrid of 5 houses whilst the remaining six cases investigate a microgrid of 10 houses. A summary of the investigated cases is given in

Table 1. Summary of the investigated cases in this paper.

CASE #	5 Houses	CASE #	10 Houses
CASE 1.0	Uncertainty Degradation	CASE 2.0	Uncertainty Degradation
CASE 1.1	10% of load uncertainty on 1000 h per year Degradation	CASE 2.1	Uncertainty (Same as CASE 1.1) Degradation
CASE 1.2	10% of load uncertainty on 2000 h per year Degradation	CASE 2.2	Uncertainty (Same as CASE 1.2) Degradation
CASE 1.3	Uncertainty Degradation	CASE 2.3	Uncertainty Degradation
CASE 1.4	Uncertainty (Same as CASE 1.1) Degradation	CASE 2.4	Uncertainty (Same as CASE 1.1) Degradation
CASE 1.5	Uncertainty (Same as CASE 2.1) Degradation	CASE 2.5	Uncertainty (Same as CASE 1.2) Degradation

. CASE 1.0 and CASE 2.0 represent the basic cases where neither uncertainty nor degradation is considered. In CASE 1.1 and CASE 2.1, uncertainty is taken into consideration, with 10% of load uncertainty at 1000 h per year. In CASE 1.2 and CASE 2.2, the same load uncertainty is considered at 2000 h per year. In CASE 1.3 and CASE 2.3, the effect of battery degradation is considered and investigated. CASE 1.4 and CASE 1.5 are similar to CASE 1.1 and CASE 1.2, and the effect of battery degradation is taken into consideration. Likewise, CASE 2.4 and CASE 2.5 are similar to CASE 2.1 and CASE 2.2 considering, in addition to the load uncertainty, the effect of battery degradation.

All simulations, modeling, and experiments are conducted using MATLAB commercial software on a personal computer with an Intel Core i7-6500U CPU @ 2.50 GHz and 5 GB RAM. The hourly load profile of a single house over a one-year timespan is generated through eQuest, a building energy simulation software [89]. The profiles, which are monthly averaged, can be found in [32], and the economic parameters of the HMS are listed in

Table 2. Economic parameters of the HMS [30].

Parameter	HMS Component
	PV	Wind	Diesel Generator	Inverter	Battery
Efficiency (%)	95 (regulator)	95 (regulator)		92	85
Lifetime (years)	24	24	3	24	12
Initial cost (USD/kW)	3400	2000	1000	2500	280 (USD/kWh)
Rated power (kW)	7.3	5	4		40 (kWh)
Regulator cost (USD)	1500	1000
Model		ZEYU FD-2KW
Cut in (m/s)		2.5
Cut out (m/s)		40
Rated speed (m/s)		9.5
Economic parameters
Discount rate (%)	Real interest (%)	Fuel inflation rate (%)	O&M running cost (%)		Project lifetime (%)
8	13	5	20		24

.

5.2. Five Houses’ Cases

The proposed IMOEAD is applied to CASE 1.0 to CASE 1.5 by assuming a population of 100 candidates over 200 iterations. The PFs obtained for CASE 1.0 are represented in Figure 8. This front is composed of 100 solutions. It can be seen from Figure 8 that the obtained PF is not smooth (it can be decomposed into four small zones), which is mainly due to the discrete nature of some variables.

The resulting fronts for CASE 1.0, CASE 1.1, and CASE 1.2 are partitioned into four zones, and a magnified view of each zone is presented separately in Figure 9. The first zone shows similar results for the three cases. However, in the remaining zones, the results are different with and without considering the issue of uncertainty. It is worth mentioning that the IMOEAD obtains 100, 102, and 95 solutions for CASE 1.0, CASE 1.1, and CASE 1.2, respectively.

The PF obtained in the case considering no degradation in the batteries (i.e., in CASE 1.0, which is composed of 100 solutions) is compared to the PF obtained in the case considering degradation in the batteries (i.e., CASE 1.3, which is composed of 106 solutions) in Figure 10. There is a clear effect of battery degradation on the quality of the solutions, which is expected in real life.

A comparison of the PFs obtained for the cases considering degradation (i.e., CASE 1.3, CASE 1.4, and CASE 1.5) is shown in Figure 11. In this figure, the PFs are decomposed into three separate zones, and for clear presentation, a zoomed-in version of each zone is presented. It can be seen from this figure that the uncertainty has very little effect on the first zone. However, this effect becomes significant in the remaining zones of the PF. Finally, it is worth mentioning that the numbers of nondominated solutions found using the IMOEAD for CASE 1.3, CASE 1.4, and CASE 1.5 are 106, 95, and 92, respectively.

Twenty selected solutions from each obtained PF for CASE 1.0, CASE 1.1, CASE 1.2, CASE 1.3, CASE 1.4, and CASE 1.5 are presented in

Table 3. Twenty selected solutions for CASE 1.0.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	43.43	4.99	9	3	0.1159	63.10	83.71	88,645.97	12,180.59	46,068.49	23,923.68
Solution # 2	33.36	4.99	10	3	0.1175	57.37	80.35	68,098.57	13,533.99	46,810.32	25,237.44
Solution # 3	33.06	5.00	10	3	0.1184	56.16	80.19	67,471.12	13,533.99	46,870.40	25,336.80
Solution # 4	43.98	4.96	9	3	0.1192	54.91	83.74	89,767.21	12,180.59	45,997.24	24,056.16
Solution # 5	20.76	4.26	9	4	0.1200	53.60	58.38	42,371.06	12,180.59	51,788.15	44,263.04
Solution # 6	44.19	4.64	1	3	0.1208	52.53	77.67	90,207.82	1353.40	52,627.11	32,192.64
Solution # 7	34.23	4.99	10	3	0.1217	48.07	80.80	69,863.69	13,533.99	46,646.39	24,972.48
Solution # 8	43.08	5.00	9	3	0.1224	47.14	83.62	87,931.77	12,180.59	46,114.35	23,945.76
Solution # 9	44.39	4.95	9	3	0.1233	46.04	83.85	90,615.16	12,180.59	45,944.16	24,023.04
Solution # 10	33.06	5.00	10	3	0.1243	44.62	80.19	67,484.73	13,533.99	46,869.09	25,336.80
Solution # 11	43.98	4.96	9	3	0.1253	42.66	83.74	89,767.21	12,180.59	45,997.24	24,056.16
Solution # 12	20.76	4.26	9	4	0.1263	41.72	58.38	42,371.06	12,180.59	51,788.15	44,263.04
Solution # 13	44.19	4.64	1	3	0.1263	37.53	77.67	90,207.82	1353.40	52,627.11	32,192.64
Solution # 14	44.49	5.00	9	3	0.1280	36.00	84.00	90,807.09	12,180.59	45,932.16	23,824.32
Solution # 15	33.06	5.00	10	3	0.1299	34.94	80.19	67,484.73	13,533.99	46,869.09	25,336.80
Solution # 16	43.43	4.99	9	3	0.1376	33.61	83.71	88,655.69	12,180.59	46,067.86	23,934.72
Solution # 17	44.82	5.00	10	3	0.1763	22.94	84.74	91,494.15	13,533.99	45,072.29	22,908.00
Solution # 18	43.43	4.99	9	3	0.1781	21.66	83.71	88,655.69	12,180.59	46,067.86	23,934.72
Solution # 19	33.06	5.00	10	3	0.1865	20.38	80.19	67,471.12	13,533.99	46,870.40	25,336.80
Solution # 20	20.76	4.26	9	4	0.2160	10.29	58.38	42,371.06	12,180.59	51,788.15	44,263.04

,

Table 4. Twenty selected solutions for CASE 1.1.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	44.66	4.99	9	3	0.1160	63.21	84.03	91,149.54	12,180.59	45,956.12	23,846.40
Solution # 2	42.69	5.00	9	3	0.1163	58.99	83.45	87,133.04	12,180.59	46,214.81	24,078.24
Solution # 3	44.55	4.94	10	3	0.1171	57.74	84.41	90,924.96	13,533.99	45,150.16	23,327.52
Solution # 4	45.00	4.98	9	3	0.1180	56.41	84.11	91,851.93	12,180.59	45,912.36	23,824.32
Solution # 5	45.00	5.00	10	3	0.1189	55.31	84.77	91,851.63	13,533.99	45,094.66	22,919.04
Solution # 6	45.00	5.00	10	3	0.1197	53.94	84.77	91,851.93	13,533.99	45,094.64	22,919.04
Solution # 7	44.53	4.99	9	3	0.1211	51.93	84.01	90,901.78	12,180.59	45,971.70	23,835.36
Solution # 8	32.93	5.00	10	3	0.1225	47.31	80.11	67,215.92	13,533.99	46,950.62	25,403.04
Solution # 9	17.65	2.21	5	4	0.1237	45.37	44.60	36,029.35	6766.99	58,415.20	56,068.48
Solution # 10	44.55	4.94	10	3	0.1243	44.54	84.41	90,924.96	13,533.99	45,150.16	23,327.52
Solution # 11	45.00	4.98	9	3	0.1249	43.41	84.11	91,851.93	12,180.59	45,912.36	23,824.32
Solution # 12	45.00	5.00	10	3	0.1254	42.37	84.77	91,851.63	13,533.99	45,094.66	22,919.04
Solution # 13	42.69	5.00	9	3	0.1262	37.59	83.45	87,133.04	12,180.59	46,214.81	24,078.24
Solution # 14	32.93	5.00	10	3	0.1278	36.06	80.11	67,215.92	13,533.99	46,950.62	25,403.04
Solution # 15	44.55	4.94	10	3	0.1296	35.10	84.41	90,924.96	13,533.99	45,150.16	23,327.52
Solution # 16	45.00	4.98	9	3	0.1384	33.52	84.11	91,851.93	12,180.59	45,912.36	23,824.32
Solution # 17	45.00	4.77	10	3	0.1766	22.58	84.01	91,851.56	13,533.99	45,094.66	24,056.16
Solution # 18	44.53	4.99	9	3	0.1787	21.19	84.01	90,901.78	12,180.59	45,971.70	23,835.36
Solution # 19	19.45	2.21	5	4	0.1865	19.99	49.01	39,698.93	6766.99	56,797.48	52,653.44
Solution # 20	44.45	5.00	10	3	0.2139	10.35	84.57	90,728.97	13,533.99	45,162.18	23,051.52

,

Table 5. Twenty selected solutions for CASE 1.2.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	44.18	4.76	1	3	0.1161	63.18	78.06	90,187.91	1353.40	52,601.02	31,618.56
Solution # 2	24.14	1.25	6	4	0.1163	59.11	54.91	49,275.02	8120.39	53,074.37	49,812.48
Solution # 3	44.19	4.76	1	3	0.1172	58.11	78.07	90,205.25	1353.40	52,599.74	31,618.56
Solution # 4	44.18	4.76	1	3	0.1181	56.40	78.06	90,187.91	1353.40	52,601.02	31,618.56
Solution # 5	41.39	5.00	10	3	0.1189	55.36	83.76	84,479.67	13,533.99	45,497.33	23,305.44
Solution # 6	44.72	4.98	9	3	0.1197	54.09	84.03	91,272.70	12,180.59	45,886.23	23,846.40
Solution # 7	44.27	4.97	9	3	0.1206	52.86	83.84	90,356.41	12,180.59	45,943.00	23,989.92
Solution # 8	33.38	5.00	10	3	0.1218	48.17	80.39	68,129.68	13,533.99	46,783.90	25,182.24
Solution # 9	33.38	5.00	10	3	0.1226	47.06	80.40	68,132.39	13,533.99	46,783.64	25,182.24
Solution # 10	24.14	1.25	6	4	0.1238	45.21	54.91	49,275.02	8120.39	53,074.37	49,812.48
Solution # 11	44.18	4.76	1	3	0.1250	43.22	78.06	90,187.91	1353.40	52,601.02	31,618.56
Solution # 12	44.36	4.98	9	3	0.1261	41.50	83.91	90,549.90	12,180.59	45,930.98	23,923.68
Solution # 13	44.27	4.97	9	3	0.1265	37.19	83.84	90,356.41	12,180.59	45,943.00	23,989.92
Solution # 14	33.38	5.00	10	3	0.1286	35.59	80.39	68,129.68	13,533.99	46,783.90	25,182.24
Solution # 15	44.19	4.76	1	3	0.1318	34.50	78.07	90,205.25	1353.40	52,599.74	31,618.56
Solution # 16	34.92	4.97	4	3	0.1393	33.25	76.54	71,271.76	5413.59	51,648.75	30,106.08
Solution # 17	24.14	1.25	6	4	0.1762	22.71	54.91	49,275.02	8120.39	53,074.37	49,812.48
Solution # 18	43.96	4.93	10	3	0.1772	21.63	84.25	89,720.80	13,533.99	45,164.60	23,371.68
Solution # 19	34.92	4.97	4	3	0.1867	19.97	76.54	71,271.76	5413.59	51,648.75	30,106.08
Solution # 20	43.06	5.00	10	3	0.2121	10.46	84.29	87,894.04	13,533.99	45,278.26	23,051.52

,

Table 6. Twenty selected solutions for CASE 1.3.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	39.28	5.00	1	3	0.1147	56.37	76.64	80,184.76	1353.40	53,423.34	31,530.24
Solution # 2	36.74	4.99	4	3	0.1157	54.21	77.65	75,000.73	5413.59	51,306.67	29,443.68
Solution # 3	35.13	5.00	1	2	0.1165	52.18	79.46	71,702.13	1353.40	54,249.22	26,150.08
Solution # 4	39.26	5.00	1	3	0.1172	49.85	76.63	80,140.24	1353.40	53,427.22	31,530.24
Solution # 5	36.54	5.00	1	3	0.1182	47.64	75.20	74,576.48	1353.40	53,948.12	32,203.68
Solution # 6	32.43	5.00	10	3	0.1185	41.86	79.76	66,185.89	13,533.99	46,996.42	25,645.92
Solution # 7	32.38	5.00	10	3	0.1193	40.23	79.72	66,082.49	13,533.99	47,006.77	25,679.04
Solution # 8	19.17	1.91	2	3	0.1200	38.81	45.51	39,125.47	2706.80	60,209.30	55,597.44
Solution # 9	36.95	4.97	1	3	0.1209	37.20	75.37	75,416.35	1353.40	53,864.27	32,181.60
Solution # 10	39.35	4.99	1	3	0.1215	35.69	76.66	80,319.01	1353.40	53,411.69	31,530.24
Solution # 11	33.21	4.92	3	3	0.1218	34.93	74.31	67,794.01	4060.20	52,921.45	32,060.16
Solution # 12	37.32	4.99	10	3	0.1233	33.60	82.17	76,176.16	13,533.99	46,116.81	24,221.76
Solution # 13	32.38	5.00	10	3	0.1250	32.96	79.72	66,082.49	13,533.99	47,006.77	25,679.04
Solution # 14	36.95	4.97	1	3	0.1275	32.37	75.37	75,416.35	1353.40	53,864.27	32,181.60
Solution # 15	37.32	4.99	10	3	0.1318	31.52	82.17	76,176.16	13,533.99	46,116.81	24,221.76
Solution # 16	36.95	4.97	1	3	0.1394	30.97	75.37	75,416.35	1353.40	53,864.27	32,181.60
Solution # 17	36.54	5.00	1	3	0.1757	14.29	75.21	74,581.65	1353.40	53,947.60	32,203.68
Solution # 18	32.38	5.00	10	3	0.1790	13.60	79.72	66,084.30	13,533.99	47,006.59	25,679.04
Solution # 19	35.13	4.99	1	3	0.2170	8.43	74.26	71,702.13	1353.40	54,249.22	32,766.72
Solution # 20	39.16	5.00	1	3	0.2487	7.67	76.59	79,936.44	1353.40	53,445.00	31,541.28

,

Table 7. Twenty selected solutions for CASE 1.4.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	36.29	4.67	0	3	0.1149	56.09	73.38	74,081.23	0.00	54,904.47	34,334.40
Solution # 2	36.56	4.97	0	3	0.1158	53.29	74.62	74,630.22	0.00	54,847.39	32,866.08
Solution # 3	41.27	4.98	1	3	0.1166	51.33	77.51	84,243.13	1353.40	53,108.81	31,199.04
Solution # 4	44.68	4.98	3	3	0.1176	49.22	80.26	91,204.87	4060.20	50,903.19	28,858.56
Solution # 5	33.02	5.00	10	3	0.1183	41.75	80.10	67,394.63	13,533.99	46,908.34	25,436.16
Solution # 6	33.02	5.00	10	3	0.1188	40.94	80.10	67,409.06	13,533.99	46,906.93	25,436.16
Solution # 7	44.45	4.98	3	3	0.1193	40.25	80.19	90,731.53	4060.20	50,936.41	28,869.60
Solution # 8	18.81	1.03	5	4	0.1199	38.88	45.62	38,386.16	6766.99	57,280.86	55,700.48
Solution # 9	34.59	4.99	0	3	0.1203	37.82	73.27	70,602.75	0.00	55,287.48	33,649.92
Solution # 10	44.73	4.97	3	3	0.1209	36.86	80.25	91,302.70	4060.20	50,896.33	28,880.64
Solution # 11	36.21	4.91	0	3	0.1213	35.74	74.23	73,910.87	0.00	54,922.35	33,197.28
Solution # 12	40.85	4.99	1	3	0.1218	34.95	77.38	83,381.78	1353.40	53,178.84	31,199.04
Solution # 13	44.66	4.98	3	3	0.1221	34.35	80.24	91,160.81	4060.20	50,906.28	28,869.60
Solution # 14	33.02	5.00	10	3	0.1233	33.47	80.10	67,394.63	13,533.99	46,908.34	25,436.16
Solution # 15	18.81	1.03	5	4	0.1250	32.93	45.62	38,386.16	6766.99	57,280.86	55,700.48
Solution # 16	40.12	4.99	1	3	0.1282	32.26	77.03	81,899.81	1353.40	53,302.55	31,364.64
Solution # 17	34.59	4.99	0	3	0.1363	31.12	73.27	70,602.75	0.00	55,287.48	33,649.92
Solution # 18	33.02	5.00	10	3	0.1780	13.97	80.10	67,394.63	13,533.99	46,908.34	25,436.16
Solution # 19	34.59	4.99	0	3	0.1927	13.15	73.27	70,602.75	0.00	55,287.48	33,649.92
Solution # 20	34.59	4.99	0	3	0.2443	7.95	73.27	70,602.75	0.00	55,287.48	33,649.92

and

Table 8. Twenty selected solutions for CASE 1.5.

Solution #	PV (kW)	AD	nWT	nDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	35.12	5.00	1	3	0.1147	56.10	74.42	71,685.37	1353.40	54,206.89	32,545.92
Solution # 2	26.93	1.94	10	3	0.1157	53.96	65.27	54,976.20	13,533.99	48,281.56	40,560.96
Solution # 3	44.27	4.97	7	3	0.1165	52.67	82.70	90,359.14	9473.79	47,553.35	25,491.36
Solution # 4	44.29	4.97	7	3	0.1175	49.49	82.70	90,399.16	9473.79	47,550.75	25,502.40
Solution # 5	37.19	4.99	1	3	0.1186	42.00	75.65	75,916.60	1353.40	53,771.17	31,905.60
Solution # 6	44.84	4.86	4	3	0.1202	38.50	80.62	91,529.58	5413.59	49,963.63	28,472.16
Solution # 7	35.42	4.99	1	3	0.1211	36.60	74.67	72,296.69	1353.40	54,140.81	32,369.28
Solution # 8	35.48	4.99	1	3	0.1217	35.20	74.69	72,421.06	1353.40	54,127.47	32,369.28
Solution # 9	32.64	5.00	10	3	0.1223	34.35	79.98	66,626.69	13,533.99	46,904.86	25,436.16
Solution # 10	32.64	5.00	10	3	0.1228	33.76	79.98	66,628.05	13,533.99	46,904.73	25,436.16
Solution # 11	26.93	1.94	10	3	0.1245	32.85	65.27	54,976.20	13,533.99	48,281.56	40,560.96
Solution # 12	44.29	4.97	7	3	0.1272	32.15	82.70	90,399.16	9473.79	47,550.75	25,502.40
Solution # 13	35.42	4.99	1	3	0.1301	31.75	74.67	72,296.69	1353.40	54,140.81	32,369.28
Solution # 14	34.49	5.00	10	3	0.1339	31.24	81.00	70,392.98	13,533.99	46,552.64	24,784.80
Solution # 15	32.64	5.00	10	3	0.1408	30.58	79.98	66,628.05	13,533.99	46,904.73	25,436.16
Solution # 16	44.29	4.97	7	3	0.1773	13.67	82.70	90,399.16	9473.79	47,550.75	25,502.40
Solution # 17	44.74	4.95	5	3	0.1919	12.88	81.55	91,326.32	6766.99	49,145.43	27,169.44
Solution # 18	31.39	5.00	10	2	0.2138	8.78	82.07	64,080.30	13,533.99	47,163.58	22,367.04
Solution # 19	32.64	5.00	10	3	0.2348	8.24	79.98	66,628.05	13,533.99	46,904.73	25,436.16
Solution # 20	42.24	4.94	1	3	0.2484	7.73	77.81	86,212.76	1353.40	52,879.45	31,165.92

, respectively. To make it easier, these solutions are arranged in order of COE results. Furthermore, some of the solutions are described herewith.

Solution #1 of CASE 1.0 is characterized by a system comprising a PV panels’ power of 43.43 kW, 5 days of autonomy, 9 WTs, and 3 diesel generators. The corresponding COE, LPSP, and RF values for this solution are 0.1159 USD/kWh, 63.10%, and 83.71%, respectively. The energy contribution of each source type, including PV panels, wind turbines, diesel generators, and batteries, can be found in in Figure 12a.

For solution # 1 of CASE 1.0, the system has a PV panels’ power of 43.43 kW, 5 days of autonomy, 9 WTs, and 3 diesel generators. This solution corresponds to a COE of 0.1159 USD/kWh, LPSP of 63.10%, and RF of 83.71%. Figure 12 presents the share of different energy resources: wind, diesel, PV, and battery.

For the # 10th solution of CASE 1.2, the system consists of 24.14 kW of PV panels, 1.25 days of autonomy, 6 wind turbines, and 4 diesel generators. The corresponding COE is 0.1238 USD/kWh, the LPSP is 45.21%, and the reserve factor (RF) is 54.91%. Figure 12c displays the energy contribution of each power source for this solution.

Another solution to be described here is solution # 5 of CASE 1.3. The system designed by this solution has a PV power of 36.5 kW, 5 days of autonomy, 1 WT, and 3 diesel generators. This gives a COE, an LPSP, and an RE of 0.1182 USD/kWh, 47.64%, and 75.20%, respectively. Furthermore, different contributions of each part for this solution are shown in Figure 12d.

In addition to the above-described solutions, two more solutions, namely, solution # 10 of CASE 1.4 and solution # 20 of CASE 1.5, are represented in Figure 12e,f, respectively.

5.3. Ten Houses’ Cases

Similar to the previous cases, the proposed IMOEAD has been tried on 10 house contained load cases to run CASE 2.0 to CASE 2.5, assuming a population size of 100 candidates over 200 iterations. The PF, which is composed of 87 nondominated solutions, obtained for CASE 2.0 is represented in Figure 13. For the same reasons mentioned before, the obtained PF is not smooth, and it is spread over five zones as it can be seen in Figure 13.

The resulting fronts for CASE 2.0, CASE 2.1, and CASE 2.2 (composed of 87, 109, and 102 solutions, respectively) are represented in Figure 14. In some zones, the results are like each other; however, in some other zones, they are quite different. However, the effect of load uncertainty is clearer in the cases of 10 houses than that of 5 houses.

The PF obtained in the case considering no degradation in the batteries (i.e., in CASE 2.0, which is composed of 87 solutions) and the one obtained in the case considering degradation in the batteries (i.e., CASE 2.3, which is composed of 140 solutions) are compared in Figure 15. It can be seen from this figure that there is a clear effect of the batteries’ degradation on the quality of the solutions like in the case of five houses.

A comparison of the PFs obtained for the cases considering degradation with or without load uncertainty (i.e., CASE 2.3, CASE 2.4, and CASE 2.5) is shown in Figure 16. It can be seen from this figure that the uncertainty has little effect on some zones and a significant impact on others. Finally, it is worth mentioning that the numbers of nondominated solutions for CASE 2.3, CASE 2.4, and CASE 2.5 are 140, 130, and 132, respectively.

For the cases related to 10 houses, twenty selected solutions from each obtained PF for CASE 2.0, CASE 2.1, CASE 2.2, CASE 2.3, CASE 2.4, and CASE 2.5 are tabulated in

Table 9. Twenty selected solutions for CASE 2.0.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	45.00	3.70	10	4	0.0716	99.99	66.12	91,851.87	13,533.99	109,654.81	72,849.28
Solution # 2	45.00	3.69	10	4	0.0759	99.55	66.12	91,848.30	13,533.99	109,655.68	72,864.00
Solution # 3	45.00	3.68	10	4	0.0780	97.86	66.12	91,849.12	13,533.99	109,655.48	72,864.00
Solution # 4	45.00	3.69	10	4	0.0912	71.77	66.12	91,848.13	13,533.99	109,655.72	72,864.00
Solution # 5	21.50	3.07	4	2	0.0916	70.46	68.13	43,874.87	5413.59	148,289.70	62,972.16
Solution # 6	15.00	1.00	0	1	0.1395	62.12	83.91	30,617.31	0.00	166,960.85	31,791.52
Solution # 7	45.00	3.57	10	4	0.1418	61.86	66.12	91,848.30	13,533.99	109,655.68	72,864.00
Solution # 8	45.00	3.43	10	4	0.1419	61.42	66.11	91,848.38	13,533.99	109,655.66	72,878.72
Solution # 9	17.72	3.04	4	2	0.1449	59.69	67.83	36,164.91	5413.59	155,999.66	63,560.96
Solution # 10	44.98	2.77	9	4	0.1908	55.51	65.50	91,818.51	12,180.59	110,650.67	74,056.32
Solution # 11	44.99	3.63	10	4	0.1925	54.63	66.11	91,823.48	13,533.99	109,661.76	72,878.72
Solution # 12	45.00	3.69	10	4	0.1934	54.24	66.12	91,848.38	13,533.99	109,655.66	72,864.00
Solution # 13	15.00	1.00	0	1	0.1989	54.05	83.91	30,617.31	0.00	166,960.85	31,791.52
Solution # 14	45.00	3.69	10	4	0.2545	43.97	66.12	91,851.90	13,533.99	109,654.80	72,864.00
Solution # 15	45.00	2.33	10	4	0.2546	43.80	65.51	91,851.93	13,533.99	109,654.80	74,159.36
Solution # 16	21.50	3.07	4	2	0.2547	43.61	68.13	43,874.87	5413.59	148,289.70	62,972.16
Solution # 17	44.97	1.44	10	4	0.2549	43.46	63.92	91,789.48	13,533.99	109,670.09	77,574.40
Solution # 18	15.00	1.00	0	1	0.2551	43.31	83.91	30,617.31	0.00	166,960.85	31,791.52
Solution # 19	45.00	3.69	10	4	0.2553	43.14	66.12	91,848.38	13,533.99	109,655.66	72,864.00
Solution # 20	45.00	3.70	10	4	0.2576	42.93	66.12	91,851.87	13,533.99	109,654.81	72,849.28

,

Table 10. Twenty selected solutions for CASE 2.1.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	45.00	3.69	10	4	0.0734	99.93	66.14	91,850.48	13,533.99	109,593.94	72,790.40
Solution # 2	44.91	3.70	10	4	0.0772	97.99	66.08	91,673.65	13,533.99	109,637.81	72,878.72
Solution # 3	44.89	3.72	10	4	0.0788	95.71	66.06	91,635.26	13,533.99	109,647.37	72,908.16
Solution # 4	45.00	3.84	10	4	0.0795	70.68	66.19	91,851.93	13,533.99	109,593.58	72,687.36
Solution # 5	45.00	3.72	10	4	0.0800	68.78	66.15	91,849.70	13,533.99	109,594.13	72,775.68
Solution # 6	45.00	3.72	10	4	0.0807	67.69	66.15	91,849.84	13,533.99	109,594.10	72,775.68
Solution # 7	45.00	2.38	10	4	0.0814	66.90	65.59	91,843.75	13,533.99	109,595.61	73,982.72
Solution # 8	45.00	3.69	10	4	0.0828	65.65	66.14	91,850.48	13,533.99	109,593.94	72,790.40
Solution # 9	22.42	2.39	4	2	0.0835	65.01	68.31	45,769.58	5413.59	146,325.07	62,596.80
Solution # 10	45.00	3.69	10	4	0.0842	64.39	66.14	91,849.82	13,533.99	109,594.11	72,790.40
Solution # 11	45.00	3.72	10	4	0.0848	63.67	66.15	91,850.36	13,533.99	109,593.97	72,775.68
Solution # 12	45.00	3.72	10	4	0.0868	62.51	66.15	91,849.70	13,533.99	109,594.13	72,775.68
Solution # 13	45.00	3.69	10	4	0.0904	61.84	66.14	91,850.48	13,533.99	109,593.94	72,790.40
Solution # 14	45.00	3.51	9	4	0.1329	60.41	65.74	91,851.93	12,180.59	110,583.09	73,526.40
Solution # 15	45.00	2.38	10	4	0.1369	59.48	65.59	91,843.75	13,533.99	109,595.61	73,982.72
Solution # 16	45.00	3.69	10	4	0.1903	55.68	66.14	91,849.82	13,533.99	109,594.11	72,790.40
Solution # 17	45.00	2.38	10	4	0.1910	54.80	65.59	91,843.75	13,533.99	109,595.61	73,982.72
Solution # 18	42.75	4.14	10	4	0.1936	54.02	64.18	87,264.25	13,533.99	110,813.31	75,793.28
Solution # 19	45.00	3.72	10	4	0.2546	43.41	66.15	91,849.70	13,533.99	109,594.13	72,775.68
Solution # 20	23.69	2.37	6	1	0.2556	42.69	84.01	48,358.58	8120.39	141,032.02	31,578.08

,

Table 11. Twenty selected solutions for CASE 2.2.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	44.99	2.45	10	4	0.0734	99.92	65.60	91,839.91	13,533.99	109,804.46	74,012.16
Solution # 2	44.88	3.60	10	4	0.0771	97.87	66.01	91,600.77	13,533.99	109,864.39	73,084.80
Solution # 3	44.88	3.60	10	4	0.0787	95.67	66.01	91,600.77	13,533.99	109,864.39	73,084.80
Solution # 4	44.97	2.32	10	4	0.0794	71.19	65.53	91,795.77	13,533.99	109,815.48	74,159.36
Solution # 5	45.00	2.32	10	4	0.0795	70.51	65.54	91,848.42	13,533.99	109,802.34	74,144.64
Solution # 6	44.88	3.55	10	4	0.0800	68.89	66.00	91,600.60	13,533.99	109,864.43	73,099.52
Solution # 7	18.80	4.12	1	2	0.0807	67.77	67.86	38,366.18	1353.40	157,996.50	63,538.88
Solution # 8	44.88	3.55	10	4	0.0813	66.89	66.00	91,597.04	13,533.99	109,865.33	73,099.52
Solution # 9	44.88	3.55	10	4	0.0820	66.26	66.00	91,596.90	13,533.99	109,865.36	73,099.52
Solution # 10	44.99	3.01	10	4	0.0827	65.61	65.99	91,833.09	13,533.99	109,806.16	73,187.84
Solution # 11	44.91	3.45	10	4	0.0834	64.97	66.04	91,674.10	13,533.99	109,845.96	73,040.64
Solution # 12	44.88	3.60	10	4	0.0846	63.77	66.01	91,600.77	13,533.99	109,864.39	73,084.80
Solution # 13	44.88	3.55	10	4	0.0867	62.50	66.00	91,600.60	13,533.99	109,864.43	73,099.52
Solution # 14	43.28	3.70	10	4	0.0911	61.63	64.62	88,334.07	13,533.99	110,723.73	75,204.48
Solution # 15	44.88	3.55	10	4	0.1327	60.54	66.00	91,600.60	13,533.99	109,864.43	73,099.52
Solution # 16	44.88	3.55	10	4	0.1353	59.83	66.00	91,597.04	13,533.99	109,865.33	73,099.52
Solution # 17	44.88	3.60	10	4	0.1906	55.10	66.01	91,600.77	13,533.99	109,864.39	73,084.80
Solution # 18	44.97	2.32	10	4	0.1928	53.99	65.53	91,795.77	13,533.99	109,815.48	74,159.36
Solution # 19	18.80	4.12	1	2	0.2544	43.56	67.86	38,365.81	1353.40	157,996.87	63,538.88
Solution # 20	44.88	3.60	10	4	0.2554	42.70	66.01	91,600.60	13,533.99	109,864.43	73,084.80

,

Table 12. Twenty selected solutions for CASE 2.3.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	44.99	4.99	10	4	0.0722	99.51	66.17	91,832.55	13,533.99	109,659.54	72,746.24
Solution # 2	44.99	4.97	10	4	0.0741	98.14	66.16	91,827.91	13,533.99	109,660.67	72,760.96
Solution # 3	39.95	4.86	10	4	0.0758	96.32	61.45	81,543.00	13,533.99	112,657.55	80,076.80
Solution # 4	44.83	5.00	10	4	0.0774	93.56	66.04	91,508.64	13,533.99	109,739.33	72,937.60
Solution # 5	44.99	4.99	10	4	0.0785	69.05	66.16	91,827.84	13,533.99	109,660.69	72,760.96
Solution # 6	45.00	2.84	10	4	0.0804	66.38	65.84	91,851.62	13,533.99	109,654.87	73,467.52
Solution # 7	44.94	5.00	10	4	0.0829	63.23	66.13	91,724.46	13,533.99	109,686.06	72,805.12
Solution # 8	44.99	4.47	10	4	0.0855	60.67	66.17	91,834.96	13,533.99	109,658.95	72,746.24
Solution # 9	45.00	2.84	10	4	0.0893	58.69	65.84	91,851.62	13,533.99	109,654.87	73,467.52
Solution # 10	44.52	4.86	10	4	0.0937	56.02	65.83	90,876.48	13,533.99	109,896.81	73,232.00
Solution # 11	45.00	4.30	10	4	0.1330	53.28	66.16	91,850.33	13,533.99	109,655.19	72,760.96
Solution # 12	21.17	1.39	7	3	0.1356	51.44	53.78	43,213.00	9473.79	144,891.37	91,311.84
Solution # 13	44.05	4.74	10	4	0.1395	49.13	65.35	89,905.59	13,533.99	110,143.77	73,997.44
Solution # 14	45.00	4.23	10	4	0.1895	46.63	66.16	91,850.63	13,533.99	109,655.11	72,760.96
Solution # 15	44.99	4.99	10	4	0.1912	44.92	66.16	91,827.95	13,533.99	109,660.66	72,760.96
Solution # 16	45.00	4.70	10	4	0.1953	41.59	66.17	91,849.99	13,533.99	109,655.27	72,746.24
Solution # 17	44.99	4.99	10	4	0.2521	34.92	66.16	91,827.82	13,533.99	109,660.70	72,760.96
Solution # 18	44.99	4.95	10	4	0.2527	32.41	66.16	91,827.95	13,533.99	109,660.66	72,760.96
Solution # 19	44.99	4.99	10	4	0.2546	30.28	66.16	91,827.91	13,533.99	109,660.67	72,760.96
Solution # 20	45.00	2.84	10	4	0.2574	27.82	65.84	91,851.62	13,533.99	109,654.87	73,467.52

,

Table 13. Twenty selected solutions for CASE 2.4.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	44.99	4.97	10	4	0.0722	99.46	66.17	91,840.97	13,533.99	109,725.41	72,775.68
Solution # 2	44.99	4.44	10	4	0.0741	97.99	66.17	91,839.72	13,533.99	109,725.71	72,775.68
Solution # 3	45.00	4.88	10	4	0.0758	96.06	66.17	91,851.89	13,533.99	109,722.71	72,775.68
Solution # 4	44.99	4.99	10	4	0.0774	93.29	66.17	91,839.72	13,533.99	109,725.71	72,775.68
Solution # 5	45.00	4.51	10	4	0.0785	69.05	66.17	91,847.06	13,533.99	109,723.90	72,775.68
Solution # 6	44.99	2.73	10	4	0.0794	67.44	65.80	91,834.75	13,533.99	109,726.94	73,555.84
Solution # 7	45.00	4.99	10	4	0.0823	64.29	66.17	91,850.76	13,533.99	109,722.99	72,775.68
Solution # 8	44.99	4.99	10	4	0.0857	60.60	66.17	91,839.72	13,533.99	109,725.71	72,775.68
Solution # 9	27.66	2.58	6	2	0.0903	58.01	70.59	56,453.84	8120.39	133,213.65	58,173.44
Solution # 10	44.99	4.99	10	4	0.0942	55.69	66.17	91,839.72	13,533.99	109,725.71	72,775.68
Solution # 11	45.00	4.99	10	4	0.1341	52.57	66.17	91,845.38	13,533.99	109,724.32	72,775.68
Solution # 12	45.00	5.00	10	4	0.1390	49.42	66.17	91,851.93	13,533.99	109,722.70	72,775.68
Solution # 13	44.99	3.20	10	4	0.1892	47.16	66.06	91,834.77	13,533.99	109,726.94	72,996.48
Solution # 14	27.66	2.58	6	2	0.1903	45.49	70.59	56,453.84	8120.39	133,213.65	58,173.44
Solution # 15	44.99	4.99	10	4	0.1920	44.04	66.17	91,840.28	13,533.99	109,725.58	72,775.68
Solution # 16	44.99	4.99	10	4	0.1960	40.97	66.17	91,839.72	13,533.99	109,725.71	72,775.68
Solution # 17	45.00	5.00	10	4	0.2521	35.31	66.17	91,843.83	13,533.99	109,724.70	72,775.68
Solution # 18	44.99	4.99	10	4	0.2526	32.84	66.17	91,839.72	13,533.99	109,725.71	72,775.68
Solution # 19	45.00	4.54	10	4	0.2551	29.88	66.17	91,850.66	13,533.99	109,723.01	72,775.68
Solution # 20	27.66	2.58	6	2	0.2577	27.64	70.59	56,453.84	8120.39	133,213.65	58,173.44

and

Table 14. Twenty selected solutions for CASE 2.5.

Solution #	PV (kW)	NAD	NWT	NDiesel	COE (USD/kWh)	LPSP (%)	RF (%)	PV (kWh)	Wind (kWh)	Battery (kWh)	Diesel (kWh)
Solution # 1	44.95	2.98	10	3	0.0705	99.90	73.34	91,751.51	13,533.99	109,688.05	57,319.68
Solution # 2	45.00	4.99	10	4	0.0740	97.97	66.16	91,845.21	13,533.99	109,664.72	72,775.68
Solution # 3	43.71	4.39	10	4	0.0758	95.94	65.06	89,228.45	13,533.99	110,340.95	74,468.48
Solution # 4	44.99	4.77	10	4	0.0775	93.33	66.15	91,839.05	13,533.99	109,666.25	72,790.40
Solution # 5	44.96	4.99	10	4	0.0785	69.16	66.14	91,772.80	13,533.99	109,682.74	72,805.12
Solution # 6	45.00	2.65	10	4	0.0804	66.42	65.75	91,848.21	13,533.99	109,663.98	73,644.16
Solution # 7	45.00	4.99	10	4	0.0813	65.21	66.16	91,845.24	13,533.99	109,664.71	72,775.68
Solution # 8	44.98	5.00	10	4	0.0832	63.13	66.14	91,808.27	13,533.99	109,673.91	72,805.12
Solution # 9	44.69	5.00	10	4	0.0856	60.53	65.90	91,211.39	13,533.99	109,824.38	73,158.40
Solution # 10	45.00	4.99	10	4	0.0902	57.79	66.16	91,845.14	13,533.99	109,664.74	72,775.68
Solution # 11	40.84	4.84	9	4	0.0938	55.71	61.97	83,358.93	12,180.59	113,124.68	79,355.52
Solution # 12	45.00	4.33	10	4	0.1331	53.09	66.15	91,847.40	13,533.99	109,664.18	72,790.40
Solution # 13	32.89	3.14	3	2	0.1370	50.57	73.70	67,123.61	4060.20	128,175.19	52,432.64
Solution # 14	44.97	4.99	10	4	0.1893	46.72	66.14	91,781.83	13,533.99	109,680.49	72,805.12
Solution # 15	44.96	4.99	10	4	0.1913	44.51	66.14	91,772.80	13,533.99	109,682.74	72,805.12
Solution # 16	32.89	3.14	3	2	0.1954	41.44	73.70	67,123.61	4060.20	128,175.19	52,432.64
Solution # 17	45.00	4.99	10	4	0.2522	35.43	66.16	91,845.18	13,533.99	109,664.73	72,775.68
Solution # 18	45.00	4.80	10	4	0.2531	31.48	66.16	91,846.38	13,533.99	109,664.43	72,775.68
Solution # 19	44.96	4.99	10	4	0.2549	29.44	66.14	91,772.80	13,533.99	109,682.74	72,805.12
Solution # 20	45.00	4.58	10	4	0.2577	27.18	66.16	91,847.51	13,533.99	109,664.15	72,775.68

, respectively. The solutions are sorted based on the values of COE, for convenience. Furthermore, some of the selected solutions are described below.

For solution # 1 of CASE 2.0, the system has a PV panels’ power of 45 kW, 3.70 days of autonomy, 10 WTs, and 4 diesel generators. The energy contribution for each source type (i.e., PV panels, wind turbines, diesel generator, and battery) for solution #1 of CASE 2.0 can be found in Figure 17a, which has a COE of 0.0716 USD/kWh, LPSP of 99.99%, and RF of 66.12%.

For solution #20 of CASE 2.1, the system consists of a PV panels’ power of 23.69 kW, 2.37 days of autonomy, 6 WTs, and 1 diesel generators. This solution has a COE of 0.2556 USD/kWh, LPSP of 42.69%, and RF of 84.01%. The energy contribution from each source type for this solution is presented in Figure 17b.

For solution # 7 of CASE 2.2, the system has a PV panels’ power of 18.80 kW, 4.12 days of autonomy, 1 WTs, and 2 diesel generators. This solution has a COE of 0.0807 USD/kWh, LPSP of 67.77%, and RF of 67.86%. The energy contribution from various resources for this solution is presented in Figure 17c.

Another solution to that described here is solution # 3 of CASE 2.3. The system designed by this solution has a PV power of 39.95 kW, 4.86 days of autonomy, 10 WT, and 4 diesel generators. This gives a COE, an LPSP, and an RE of 0.0758 USD/kWh, 96.32%, and 61.45%, respectively. Furthermore, different contributions of each part for this solution are shown in Figure 17d.

In addition to the above-described solutions, two more solutions, namely, solution # 9 of CASE 2.4 and solution # 13 of CASE 2.5, are represented in Figure 17e,f, respectively.

6. Conclusions

In this paper, a multi-objective optimization problem considering LPSP and COE was formulated to determine the optimum size of a small-scale PV/Wind/Diesel HMS system for Yanbu, Saudi Arabia, while accounting for impacts of load uncertainty and battery deterioration. An improved approach based on the MOEA/D algorithm was proposed, implemented, and applied to solve the formulated problem. The experimental investigation included ten instances representing different scenarios for five and ten houses. For the five houses cases, CASE 1.0, CASE 1.1, CASE 1.2, CASE 1.3, CASE 1.0, and CASE 1.5, 100, 102, 95, 106, 95, and 92 solutions were obtained, respectively. In these cases, the PF could be divided into four different zones. An example of solutions for CASE 1.0 is a system with a PV panels’ power of 43.43 kW, 5 days of autonomy, 9 WTs, and 3 diesel generators corresponding to a COE of 0.1159 USD/kWh, LPSP of 63.10%, and RF of 83.71%. For the ten houses’ cases, CASE 1.0, CASE 1.1, CASE 1.2, CASE 1.3, CASE 1.0, and CASE, 1.5, 87, 109, 102, 140, 130, and 132 solutions were obtained, respectively. In these cases, the PF could be divided into five different zones. An example of a solution for CASE 2.0 is a system with a PV panels’ power of 45 kW, 3.70 days of autonomy, 10 WTs, and 4 diesel generators, which has a COE of 0.0716 USD/kWh, LPSP of 99.99%, and RF of 66.12%. Furthermore, this study found that battery degradation had a significant impact, while the effect of load uncertainty was moderate in some instances and heavy in others. The obtained PF or the set of nondominated solutions covered a wide range in each case. These results can aid the HMS designer/engineer in selecting the best solution, considering other factors or constraints that cannot be modeled mathematically.