*2.1. Photovoltaic System (PV)*

A simplified model based into account ambient temperature and solar irradiation is employed in this study to compute the energy generated by the PV panels PVp(t), as indicated in the equation below [28,29].

$$\mathrm{PV\_p(t)} = \mathrm{N\_{PV}} \times \mathrm{PV\_{rat}^P} \left(\frac{\mathrm{R\_{int}(t)}}{\mathrm{R\_{STC}}}\right) \left[1 + \gamma\_\mathrm{T} \left( (\mathrm{R\_{int}(t)} \left(\frac{\mathrm{T\_{nor} - 20}}{0.8}\right) + \mathrm{T\_{amb}(t)}) - \mathrm{T\_{STC}} \right) \right] \cdot \eta\_\mathrm{w\ }\eta\_\mathrm{IV} \tag{1}$$

where, PV<sup>P</sup> rat indicates the rated power of the PV panel at standard test condition (STC) [kW], Rint(t) is the intensity of solar radiation at time (t), RSTC denote the intensity of solar radiation at standard conditions [1000 W/m2], NPV is the number of PV units, γ<sup>T</sup> is the PV module temperature coefficient [%/°C], η<sup>w</sup> is the wiring efficiency, ηPV is the PV module efficiency, Tnor is the cell temperature under at normal operating conditions, Tamb(t) denote the ambient temperature (◦C), TSTC denote the cell temperature under standard operating conditions (◦C). The technical specifications of the PV panel modeling are shown in Table 1.

**Table 1.** The main parameters of the selected photovoltaic model [30].


### *2.2. Wind Turbine*

Every month, NASA supplies data on wind speed, which has been utilized as input data for this study (NASA, 2020). The following mathematical formulas are employed to calculate the wind turbine output WTP(t) based on a comprehensive literature review [2,31].

$$\text{WT}\_{\text{P}}(\mathbf{t}) = \begin{cases} 0, & \text{V}(\mathbf{t}) < \text{V}\_{\text{cut}}^{\text{in}} \text{ or } \text{V}(\mathbf{t}) > \text{V}\_{\text{cut}}^{\text{off}} \\ \text{N}\_{\text{WT}} \times \text{WT}\_{\text{rat}}^{\text{P}} \times \text{WT} \left(\frac{\text{V}^{2}(\mathbf{t}) - \text{V}\_{\text{cut}}^{2}}{\text{V}\_{\text{rat}}^{2} - \text{V}\_{\text{cut}}^{2}}\right), & \text{V}\_{\text{cut}}^{\text{in}} < \text{V}(\mathbf{t}) < \text{V}\_{\text{rat}} \\ \text{N}\_{\text{WT}} \times \text{WT}\_{\text{rat}}^{\text{P}} \times \eta\_{\text{WT}}, & \text{V}\_{\text{rat}} < \text{V}(\mathbf{t}) < \text{V}\_{\text{cut}}^{\text{off}} \end{cases} \tag{2}$$

In which, V(t), Vin cut, Voff cut, and Vrat are WT speed at time t, WT speeds cut-in, WT speeds cut-off wind speed, and rated speed respectively. NWT denotes the number of WTs modules, ηWT is the WT efficiency, and WTP rat is the rated power of the WT (kW).

Wind speed increases with height above ground level, and the wind turbine hub's height has also a major impact on wind speed, which affects power generation, according to the below power law equation [13]:

$$\left(\frac{\text{V}\_{\text{n}}}{\text{V}\_{\text{ref}}}\right) = \left(\frac{\text{H}\_{\text{n}}}{\text{H}\_{\text{ref}}}\right)^{\varepsilon\_{\text{net}}}\tag{3}$$

where, Vn represents the WT speed (m/s) at the new height Hn (m), Vref is the WT speed (m/s) at the original turbine hub height Href (m), and εwt denotes the WT friction coefficient.

According to the International Electro technical Committee (IEC), the value of the coefficient of friction in the case of normal wind conditions is 0.20 and in the case of intensive wind conditions is 0.11. The technical specifications of the selected WT modeling are presented in Table 2.


**Table 2.** The main parameters of the selected wind turbine model [30].

### *2.3. Biomass System*

Biomass comprises of the stored chemical energy from solar energy, so biomass can be used for heating by direct burning or transformed through many operations into liquid fuels and renewable gases [32,33]. One of the major aspects in determining the type of technology used to generate biomass energy is the type of biomass to be used and the type of fuel to be produced from the conversion process [34].

In this work, biomass gasification is the conversion process used which is a pyrolysis process in which the raw materials of biomass are heated in closed and pressurized vessels, the output gaseous fuel by this process is usually called the producer gas [35].

In this study, sugarcane bagasse was used as a raw material for biomass to feed a small-scale downdraft gasifier, as the cane crop is one of the agricultural crops available in the New Valley city. The biomass generator was utilized as the primary generator to satisfy the electrical load requirement beside the PV and WT systems, the technical characteristics of the biomass system are illustrated in Table 3. The hourly generated power from the biomass system BGP(t) can be expressed according to the following mathematical formula [25,27];

$$\text{BG}\_{\text{P}}(\mathbf{t}) = \text{FS}\_{\text{rat}}(\mathbf{t}) \times \text{HHV}\_{\text{fs}} \times \eta\_{\text{gas}} \times \omega \tag{4}$$

where, FSrat(t) is the biomass raw material rate per hour (kg/h), HHVfs indicates the higher heat value of the biomass raw material, ηgas denotes the efficiency of the gasifier reactor (75%), and ω represents a factor for converting units from kJ to kWh (27.78 × 10<sup>−</sup>5).

The load ratio of the considered biomass generator is set to operate at no less than 30% (Genmin = 30%) of its rated capacity to avoid running at much lower demands, while its maximum load is 80% (Genmax = 80%) of its rated capacity. The generator output power (Genout) can be described according to the following constraints [25,27,36];

$$\text{Gen}\_{\text{out}} = \begin{cases} 0 & \text{BGp} < \text{Gen}\_{\text{min}} \\ & \text{BGp} \text{ Gen}\_{\text{max}} > \text{Gen}\_{\text{max}} \\ & \text{BGp} \text{ Gen}\_{\text{max}} < \text{BGp} < \text{Gen}\_{\text{min}} \end{cases} \tag{5}$$

**Table 3.** The main parameters characteristics of the biomass system.


Based on the previous mathematical expressions, FBG con(t) is the average fuel consumption per hour, and EBio (kWh) is the annual energy output which can be computed as following;

$$\mathop{\rm E\_{Bio}}\_{\mathbf{t}\text{-norm}} = \sum\_{t=1}^{8760} \mathop{\rm N}\_{\mathbb{R}} \times \mathbf{Gen}\_{\text{out}} \times \mathbf{t} \tag{6}$$

where, Ng is the number of generators.

### *2.4. Battery Bank Model*

The battery bank serving as a backup system of storing energy in the event that the renewable sources are unable to deliver the needed power. The hourly total power generated by the PV, WTs, and biomass system Pre(t) is obtained based on the below equation [19,25];

$$\mathbf{P\_{re}} = \mathbf{P\_{PV}} + \mathbf{W}\mathbf{T\_P} + \mathbf{B}\mathbf{G\_P}/\eta\_{\text{inv}}\tag{7}$$

The technical specifications of the battery bank model are illustrated in Table 4. The following equations explain the energy production and consumption of the battery system from time t–1 to time t [30,38];

During the charging phase BatCH,

$$\text{Bat}\_{\text{CH}}(\text{t}) = \left(\text{P}\_{\text{re}}(\text{t}) - \left(\text{P}\_{\text{L}}\left(\text{t}\right)/\text{\textdegree\_{inv}}\right)\right) \times \Delta \text{t} \times \text{\textdegree\_{CH}}\tag{8}$$

SOCBat(t)= SOCBat(t − 1) × (1 − σ) +BatCH(t) (9)

During discharging phase BatDIS,

$$\text{Bat}\_{\text{DIS}}(\mathbf{t}) = \left( \left( \mathbf{P}\_{\text{L}} \left( \mathbf{t} \right) / \eta\_{\text{inv}} \right) - \mathbf{P}\_{\text{re}} \left( \mathbf{t} \right) \right) \times \Delta \mathbf{t} \times \eta\_{\text{DIS}} \tag{10}$$

$$\text{SOC}\_{\text{Bat}}(\mathbf{t}) = \text{SOC}\_{\text{Bat}}(\mathbf{t} - \mathbf{1}) \times (1 - \sigma) - \text{Bat}\_{\text{DIS}}(\mathbf{t}) \tag{11}$$



In which, ηCH and ηDIS indicate the battery charging and discharging efficiencies, respectively, σ is self-discharge rate, and SOCBat is the battery state of charge. ηinv denotes the inverter efficiency.

### *2.5. Bi-Directional Converter Model*

A bidirectional transducer is adopted to maintain power flow between DC and AC components. There are two kinds of power conversion devices in a power system, the inverter which converts DC current to AC current and the rectifier which converts AC current to DC current. The technical characteristics of the inverter model are presented in Table 5. The hourly input power of the inverter Pinv(t) can be expressed as below [16];

$$P\_{\rm inv}(\mathbf{t}) = P\_{\rm L}(\mathbf{t}) / \eta\_{\rm inv} \tag{12}$$

In which, ηinv represents the inverter efficiency.


**Table 5.** The main parameters characteristics of the inverter.

### **3. Description of the Studied Area**

The considered area for this study is Alrashda village, which is located 10 km northwest of Mut town, the administrative center of the Dakhla Oasis in the New Valley Governorate in Egypt, at 28.938◦ east longitude, 25.576◦ north latitude, and an altitude of 243 m. The reason of choosing this village because of its comparatively high solar, wind, and biomass energy potential. The proposed mathematical model is used for designing a small scale stand-alone hybrid system to feed a range of loads which are represented in residential loads, where the peak loads are occurred during the summer and in the evening period from 19:00 to 23:00 p.m. In Figure 2, the profile of the proposed loads during a year is depicted, which shows that the average residential load of the village has reached about 260 kW, with a maximum load of 410 kW. Figures 3–5 illustrate the plot of hourly data of the solar radiation, temperature, and wind speed which are obtained from the NASA Surface Meteorology and Solar Energy website for 20 years for the selected area. Figure 3 presents the short-wave solar irradiance of the studied area during a year, where the yearly radiation rate is between 2.45 kWh/m2/day to 10.94 kWh/m2/day, with the average yearly radiation on this site's horizontal surface is around 6.89 kWh/m2/day, while the yearly ambient temperature of the selected site is indicated in Figure 4, which showed that the maximum ambient temperature can be reached, is 40◦. Figure 5 illustrates the annual wind speed for the selected location with a maximum wind speed of about 13.9 m/s and an average in the range from 8.71 m/s to 9.89 m/s. As previously mentioned, the biomass feedstock used in this study was the sugarcane bagasse. The sugar cane crop is considered one of the strategic crops in Egypt, where the harvest period begins during January of each year and extends until May. The amount of biomass feedstock available at the selected site was assumed to have a variable values over the year, the monthly biomass consumption rate is presented in Figure 6, with an average of one ton/day.

**Figure 2.** Load profile of the studied area.

**Figure 3.** The annual short-wave solar irradiance of the studied area.

**Figure 4.** The yearly ambient temperature of the studied area.

**Figure 5.** The yearly wind speed of the studied area.

**Figure 6.** The biomass consumption of the studied area.

### **4. Optimization Problem**

The primary aim of this work is to indicate the capacity to optimize the suggested stand-alone hybrid power system in order to provide a guaranteed supply of power at the lowest feasible cost. In this section, the economic and cost analysis, the main objective function, the optimization constraints, and system management strategy are discussed.

### *4.1. Economic and Cost Analysis*

The COE for a specific system is an economic evaluation of the system's costs and of the associated cost in its lifespan. The COE is a function of the NPC, it actually helps to select the lowest energy prices from different feasible hybrid configurations, which means the least overall investment cost in a renewable power system plant, after fulfilling the energy dependability limitations. While the NPC represents the current value of the capital investment and operating costs over the lifespan. The NPC and the COE in (\$/kWh) can be computed as follows [3,30]:

$$\text{COE} = \frac{\text{NPC}}{\sum\_{1}^{8760} P\_{\text{L}}} \text{C}\_{\text{RF}} \tag{13}$$

$$\text{NPC} = \text{C}\_{\text{Ann}}^{\text{T}} / \text{C}\_{\text{RF}} \tag{14}$$

where, CT Ann is the total annual cost of the proposed hybrid system, and CRF is the capital recovery factor, which is a ratio for the current cash value calculation and it can be estimated over a lifespan of years (S = 25 years) and an interest rate (Ir = 6%). CRF and C<sup>T</sup> Ann are modeled as:

$$\text{RF } (\mathbf{I}\_{\mathbf{r}} \text{ 'S}) = \frac{\mathbf{I}\_{\mathbf{r}} \times (\mathbf{I}\_{\mathbf{r}} + \mathbf{1})^{\mathbf{S}}}{\left(\mathbf{I}\_{\mathbf{r}} + \mathbf{1}\right)^{\mathbf{S}} - \mathbf{1}} \tag{15}$$

$$\mathbf{C}\_{\text{Ann}}^{\text{T}} = \sum \mathbf{C}\_{\text{Ann}}^{\text{u}} = \mathbf{C}\_{\text{Ann}}^{\text{FV}} + \mathbf{C}\_{\text{Ann}}^{\text{WT}} + \mathbf{C}\_{\text{Ann}}^{\text{BG}} + \mathbf{C}\_{\text{Ann}}^{\text{Bat}} + \mathbf{C}\_{\text{A}}^{\text{inv}} \tag{16}$$

$$\mathbf{C}\_{\text{A}}^{\text{u}} = \mathbf{C}\_{\text{Ann\\_Cap}}^{\text{u}} + \mathbf{C}\_{\text{OM}}^{\text{u}} + \mathbf{C}\_{\text{Ann\\_Rep}}^{\text{u}} + \mathbf{C}\_{\text{Ann\\_fuel}} \tag{17}$$

where, Cu Ann is the annual cost of each unit, <sup>C</sup><sup>u</sup> Ann\_Cap is the total annualized cost of each unit, Cu OM is the operation and maintenance cost of each unit, Cu Ann\_Rep is the replacement cost for each unit, and CAnn\_fuel is the annual fuel cost of the biomass unite which is computed by applying the following formula [25,27]:

$$\mathbf{C}\_{\text{Ann\\_fuel}} = \mathbf{C}\_{\text{Bio}} \times \text{Bio}\_{\text{T}} \tag{18}$$

BioT <sup>=</sup> ∑<sup>8760</sup> <sup>1</sup> FSrat(t) (19)

where, CBio is biomass fuel cost, and BioT is the total feedstock consumption of the generator (kg/year).
