# Development of Optimal Tilt Angle Models of a Photovoltaic Module for Maximum Power Production: Ethiopia

Authors: Ahunim Abebe Ashetehe; Belachew Bantyirga Gessesse; Fekadu Shewarega
Journal: International Journal of Photoenergy
 (2022)

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## Abstract

The power generated from the photovoltaic module is directly related to the magnitude of total incident solar radiation on the surface of the solar module. The total incident solar radiation depends on the location, tilt angle, and orientation of the solar module. In this paper, generic models were developed that determine the seasonal and annual optimal tilt angle of the Photovoltaic module at any location in Ethiopia without using meteorological data. Both isotropic and anisotropic diffuse solar radiation models were used to estimate monthly, seasonal, and annual optimal tilt angles. The monthly average daily global horizontal solar radiation for a total of 44 cities -32 for training and 12 for testing were obtained from the National Aeronautical and Space Administration database, and algorithms were developed and implemented using MATLAB and R programming software to obtain optimum tilt angle and regression models. The study showed that the developed model accurately estimates the optimal tilt angle with the minimum statistical validation errors. It is also found that 5.11% to 6.275% (isotropic) and 5.72% to 6.346% (anisotropic models) solar radiation energy is lost when using the yearly average fixed optimal tilt angle as compared with the monthly optimal tilt angle. The result of this study was also validated by comparing it with the previously published works, PVGIS and PVWatt online software. The graphical abstract is included in the supplementary file.


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## Body

## 1. Introduction

Ethiopia is located in the Horn of Africa at coordinates of 9.1450oN and 40.4897°E with a population of more than 114 million people, of which 78.1% live in rural areas [1]. Currently, the country’s installed grid power generation installed capacity is 4244 MW, almost entirely dependent on hydropower, which accounts for 89.9%. The remaining 7.6% and 2.5% are generated from wind and thermal sources, respectively [2]. According to the World Bank data in 2019 only 48.3% of the total population has access to electricity. From this 92.8% of the urban and 36.3% of the rural population is electrified [3]. To increase the electricity access rate, the Ethiopian government planned to increase the generation by constructing new power plants from hydropower, wind, solar, and geothermal energy sources. Among these, solar energy is one of the most environmentally friendly and promising sources of electrical power. The estimated exploitable potential ranges from 4 to 6 kWh/m2/day [4].The amount of solar energy on the surface of the photovoltaic (PV) module is affected by global horizontal solar radiation, tilt angle, orientation of the solar panel, and ground reflectance [5]. The available solar radiation at a specific site depends on the location and time of the year. Measured total solar radiation at the horizontal and at the inclined surface are not available for different sites in Ethiopia. Therefore, it is necessary to calculate the solar radiation on the surface of the module for proper design and installation of the photovoltaic systems.Maximum daily solar radiation can be obtained through the proper installation of the PV panel by optimizing the solar panel installation, tilt angle, and orientation. In general, the solar panel must be oriented toward the equator, which indicates that installations in the southern hemisphere must be oriented towards the north, and the location of Ethiopia being in the northern hemisphere, PV panels must be oriented towards the south [6]. On the other hand, the solar module installation tilt angle is influenced by the location and path of the sun. Therefore, the tilt angle can be set to one optimal value or seasonally varied to a different optimal tilt angle to obtain a maximum incident solar radiation on the surface of the solar module. Different authors have proposed the optimal tilt angle-latitude relation [7–9], as a rule of thumb formula to determine the optimal tilt angle for many locations. This method is not universally applicable and may not be an optimal tilt angle value for different locations, as the solar radiation not only varies with latitude but also varies with a change in elevation of the location. Therefore, the accurate optimum tilt angle for a certain region must be evaluated based on the location of solar radiation data.The monthly average daily total solar radiation on the surface of the solar panel (tilted surface) is the sum of beam (direct) radiation, diffuse radiation, and reflected radiation. Several models are developed by different authors to estimate the solar radiation on the inclined surface. All models apply the same principle to estimate the reflected and beam radiation. However, they are different in determining the diffuse portion of the total solar radiation incident on the inclined surface. The diffuse radiation is assumed to be uniformly distributed over the skydome (isotropic), or it is assumed to be the sum of the circumsolar region and isotopically distributed portion (anisotropic) [10].Many researchers use isotropic [11–16], anisotropic [17–19], or both isotropic and anisotropic models [5, 20–22] to predict total solar radiation on an inclined surface. Nicolás-Martín [23] proposed models that determine the annual optimal tilt angle without local meteorological data. Parameters such as latitude, albedo, and diffuse fraction were considered for the development of the optimal tilt angle models. A global optimal tilt angle regression model as a function of latitude was proposed with 2o RMSE for latitudes of -50o to 90o. The results of this literature reveal a 1% energy loss with a 10o variation of the tilt angle from the suggested optimal tilt angle values.Kallioğlu et al. [24] proposed a regression model to obtain the optimal tilt angle for three provinces in Turkey. In this literature, the highest optimal tilt angle value is found in autumn and winter and the lowest is observed in spring and summer. For the three provinces of Turkey, the optimum tilt angle model is expressed as a function of the declination angle, while for the Northern hemisphere the optimal tilt angle is determined by taking the latitude as the main parameter. The accuracy of the proposed models is compared using the coefficient of determination (R2) and the best model for provinces and the northern hemisphere obtained with R2 values of 0.9979 and 1, respectively. The authors of this work assume that the diffuse radiation is uniformly distributed across the skydome (isotropic model only). The result showed that the productivity of the monthly optimal tilt angle is increased by 16.97%, 15.57%, and 15.02% in the three provinces of Turkey as compared to the annual.Memon et al. [25] performed a case study at Sukkur IBA University Pakistan to determine the optimal tilt angle of a 1-MW photovoltaic system. The optimal tilt angle of the existing system, i.e. 15o, is compared with the tilt angle varied between 0 and 90o. The optimal tilt angle for the photovoltaic system at this location is found to be 29.5 degrees. The anisotropic model by Reindel et al. is used to determine the diffuse radiation. However, the proposed optimal tilt angle could be used only for the specified location but not for the locations that have different latitudes.Benghanem [5], conducted a case study on determining of optimal tilt angle for Madinah, Saudi Arabia. The result of the study indicated that the annual optimal tilt angle (23.5o) is nearly equal to the latitude of the study location. This fixed annual tilt angle resulted in an 8% energy loss reduction compared with the energy obtained by setting the solar panel at its monthly optimal tilt angle. Hailu et al. [20], conducted a study in Greater Toronto-Canada to obtain the optimum tilt angle and orientation of the solar panel using isotropic and anisotropic models. The study indicated that the optimum tilt angles were 37 to 44o (isotropic models) and 46 to 47o (anisotropic models) oriented west or east of due south. A 1% total solar radiation reduction was observed with a 15o change in orientation west or east of due south. The effect of the orientation of the solar panel on the outlet temperature was also investigated.Jamil et al. [16] conducted a case study in the Humid subtropical climatic region of India. In this study estimation of solar radiation and optimal tilt angle for Aligarh and New Delhi were observed. The annual optimum tilt angle for Aligarh and New Delhi was found to be 27.62o and 27.95o, respectively (close to the latitude of the respective location). The seasonal optimal tilt angle results in an energy loss of 1.16% for Aligarh and 1.18% for New Delhi. Furthermore, the annual optimal tilt angle results in an energy loss of 5.68% for Aligarh and 4.91% for New Delhi.Hassan et al. [26] conducted a case study in Iraq. The optimum tilt angle has been determined for eight cities in Iraq using an anisotropic HDKR model. The optimization process is performed by using hourly experimental solar radiation data. The results demonstrate that the maximum solar radiation can be collected with the tilt angle from 0o to 64o. The optimum tilt angle values increased during winter and decrease during summer, marking the highest values in January and December and the lowest values in June and July for all cities.In general, the optimum tilt angle of the solar module is one of the parameters that affect the output of the PV system. The summary of the reviewed literature work is given in Table1. Most of the previous research works were done for specific locations because of the site-specific parameters used. As a result, the optimal tilt angle obtained in one location may not be an optimal value for another. In references [16, 23–26] to evaluate the diffuse radiation only isotropic or anisotropic models are used. In other works [5, 20, 25] numerical optimal tilt angle values are suggested for the specific study area. These values cannot be taken as optimal values for other locations of different latitudes and elevations. Additionally, there is limited data available on determining the optimal tilt angle and the corresponding total solar radiation on the surface of the solar module for different locations in Ethiopia that could justify the optimal design of the PV system.Table 1
Summary of related previous works.
ReferencesModelAdvantagesDisadvantages[23]βopt=−0.007021ϕ2+1.091ϕ+2.132(i) Latitude, diffuse fraction, and albedo parameters are considered (ii)anisotropic (peers model) was used(i) Other isotropic and anisotropic models were not considered in the study(ii) this model is not accurately applied to other location[24]0.8845ϕ+1.5908(i) the main parameters are declination angle and latitude(ii) simple for analysis due to the consideration of the isotropic model(i) Seasonal variation of optimal tilt angle not considered(ii) the diffused solar radiation was assumed isotropic (only isotropic model is used)[25]The optimal tilt angle proposed is 29.5 degree(i) the optimal tilt angle is obtained for the study site(ii) Reindel et al. model is used(i) Other diffuse radiation evaluation techniques were not considered(ii) the result of this study can not be equally applied as optimal value for different locations[5]The optimal tilt angle is 23.5o(i) the annual optimal tilt angle found is nearly equal to the study site latitude(i) this optimal tilt angle could not be equally applied to a different latitude locations[20]The optimal tilt angles are 37 to 44o (isotropic models) and 46 to 47o (anisotropic models)(i) the study location optimal tilt angles were found using both isotropic and anisotropic models(i) the result is only applicable to the study site, no general model is suggested for other locations(ii) no comparison of the result obtained by different models using statistical tools[16]Optimal tilt angles for Aligarh and New Delhi are 27.62o and 27.95o, respectively(i) Annual optimal tilt angles of the study sites were determined(ii) A simplified isotropic model was used(iii) numerical values of the tilt angle were suggested(i) Diffused radiation was assumed uniformly distributed(ii) can not be accurately applied to other locations[26]Linear, second-order, and third-order polynomial models are proposed(i) the optimal tilt angle was obtained by using hourly experimental solar radiation data(ii) optimal tilt angles and the corresponding models were developed using an anisotropic model(i) Other parameters such as latitude were not considered in the studyThe main aims of this work are to determine the optimal tilt angle and to develop a generic optimal tilt angle model that accurately estimates the seasonal and yearly optimal tilt angle at any location in Ethiopia using different isotropic and anisotropic models without using the meteorological data.The monthly average daily global, diffuse, and beam solar radiation on a horizontal surface for 44 cities in Ethiopia were determined. Based on the maximum solar radiation received on the surface of the photovoltaic system, the monthly, seasonal, and yearly optimal tilt angles were obtained using isotropic and anisotropic models of diffuse solar radiation. The generic seasonal and yearly optimal tilt angle models as a function of latitude have been developed that accurately estimate the optimum tilt angle at any location in Ethiopia. The accuracy to predict the proposed models using both isotropic and anisotropic models has been validated using statistical indices of RMSE, MBE, MAE, MAPE, R2, previous works in the literature, and online software (PVGIS and PVWatt).The novelty of this study is: (1) solar radiation at the surface of the photovoltaic module is evaluated and determined by considering isotropic and anisotropic diffuse radiation. (2) a new and generic optimal tilt angle model as a function of the latitude, using isotropic and anisotropic diffuse radiation is developed that accurately estimates the optimal tilt angle at any location in Ethiopia without using meteorological data. (3) The models are developed and compared to use in isotropic and anisotropic diffuse radiation and the best model which accurately predicts the optimal tilt angle is selected. This model is very useful for researchers and designers in the field of photovoltaic engineering to determine and apply the optimal tilt angle model by knowing only the location latitude without using meteorological data and the declination angle at any location.The rest of the paper is organized as follows: Section2 describes the materials and methodology used in the study, i.e. used data sets, solar radiation models, and procedures for the determination of the optimal tilt angle. The result and discussion part of the paper is given in section 3 and finally, the conclusion is provided in section 4.


## 2. Materials and Methods

The solar tracker positions the solar module in the direction of the sun throughout the day to obtain the maximum solar radiation. However, this method has a high cost and requires a more complex system than the fixed tilt angle method. Therefore, it is necessary to determine the optimal tilt angle of the solar module which can be set to one optimal tilt angle value or that can be varied seasonally. Studies showed that the solar radiation on the surface of the panel varies with the change in the day of the year and latitude of the location. This phenomenon will be verified in this study. This section describes the approach used for determining the optimal tilt angle and the corresponding generic models to obtain the optimal tilt angle at any location in Ethiopia.


### 2.1. Data Set

To develop the generic optimal tilt angle models, 44 cities of Ethiopia are randomly selected and their monthly average daily global horizontal radiation was obtained from the National Aeronautical and Space Administration (NASA) database from 01/01/2010 to 31/12/2020 as shown in Figure1. A sample monthly average daily total horizontal solar radiation data for Bahir Dar city is shown in Table 2. The monthly average extraterrestrial radiation, clearness index, and diffuse radiations are evaluated for each study site using the solar radiation model and is discussed in the following subsection.Figure 1
Study sites.Table 2
Geographical coordinates and monthly average daily global horizontal radiation data of Bahir Dar city.
Latitude (o)Longitude (o)MonthGlobal horizontal radiation, H (kWh/m2/day)11.574237.3614January6.131February6.585March6.797April6.860May6.186June5.846July5.021August4.998September5.833October6.186November5.922December5.975


### 2.2. Solar Radiation on a Horizontal Surface

The solar radiation above the earth’s atmosphere is called extraterrestrial radiation. The daily extraterrestrial radiation on a horizontal surface can be calculated using equation (1) [7]
(1)Ho=24Gscπ1+0.033cos360n365cosϕcosδcosωs+πωs180sinϕsinδWhereωs is the sunset hour angle (degree) and computed as [7]:
(2)ωs=cos−1−tanϕtanδThe declination angle (δ) is the angle formed between the line joining the center of the earth and the sun and the equatorial plane. Its angle value changes over the year from 23.45o north to 23.45o south. The declination angle as a function of the day of the year (n), can be evaluated using the equation (3) [25]:
(3)δ=23.45osin360o284+n365Only a portion of extraterrestrial radiation reaches the earth’s horizontal surface, which is expressed using the clearness index (Kt). The monthly average daily clearness index is the ratio of the monthly average daily global radiation on a horizontal surface (H) to the monthly average daily global extraterrestrial radiation (Ho) and is evaluated as [11].
(4)Kt=HHoThe diffused radiation (Hd) on the horizontal surface is a function of the clearness index and global horizontal radiation and can be determined using equations (5) and (6) [27].Forωs≤81.4° and 0.3≤Kt≤0.8(5)Hd=H1.391−3.560Kt+4.189Kt2−2.137Kt3Forωs>81.4° and 0.3≤Kt≤0.8(6)Hd=H1.311−3.022Kt+3.427Kt2−1.821Kt3


### 2.3. Total Solar Radiation on a Tilted Surface

The monthly average daily total solar radiation on the inclined surface of the solar panel is composed of (1) beam or direct radiation, (2) diffuse radiation, and (3) reflected radiation [20]. Thus for the tilted surface, the incident total monthly average daily total radiation, HT (kWh/m2/day) is obtained using equation (7) [17].
(7)HT=HB+HD+HR


#### 2.3.1. Beam Radiation (HB)

The total daily beam radiation on the tilted surface of the solar energy converter can be expressed as [12].
(8)HB=H−HdRbWhere Rb is the ratio between the monthly mean daily direct radiation on a horizontal surface to the inclined surface. Rb is defined as [7]:
(9)Rb=cosθcosθzThe cosine value of incident angle, cos (θ) can be defined as a combination of different solar angles as follows [28].
(10)cosθ=sinδsinϕcosβ−sinδcosϕsinβcosγ+cosδcosϕcosβcosω+cosδsinϕsinβcosγcosω+cosδsinβsinγsinωThe cosine of the zenith angle (cos (θz)) is computed by setting the panel tilt angle to zero (β =0), as [29].
(11)cosθz=cosϕcosδcosω+sinϕsinδHence, for the south-facing surface where the azimuth angle value is zero (γ =0o), the equation (7) can be written as [30]:
(12)Rb=cosϕ−βcosδsinωss+π/180ωsssinϕ−βsinδcosϕcosδsinωs+π/180ωssinϕsinδWhereωss is the sunset hour angle for the tilted surface for the mean day of the month, obtained by equation (13) [30].
(13)ωss=mincos−1−tanϕtanδcos−1−tanϕ−βtanδ


#### 2.3.2. Diffuse Radiation (HD)

Different studies have shown that there are several empirical models to determine diffuse radiation. The sky diffuse radiation on the inclined surface of the panel can be determined by equation (14) [24]:
(14)HD=HdRdThe ratio of the average daily diffuse solar radiation on the tilted surface to the diffuse radiation on the horizontal surface, (Rd), can be evaluated by using different isotropic (equations (15)–(18) and anisotropic (equations ((19) (20) (22) and (23)) empirical formulas as followsIsotropic model: Isotropic models assume that the diffuse sky radiation is uniformly distributed over the skydome. Different researchers suggest different empirical formulas to calculate Rd, these are:Liu and Jordan’s model [31].
(15)Rd=1+cosβ2Badescu model [13].
(16)Rd=3+cos2β4Tian et al. model [14].
(17)Rd=1−β180Koronakis model [15].
(18)Rd=132+cosβAnisotropic model: The anisotropic model assumes that diffuse sky radiations are the sum of anisotropically distributed diffused radiation components in the circum-solar region (sky near the solar disc) and isotropically distributed diffuse components from the rest of the skydome.Reindl et al. model [22].
(19)Rd=HbHoRb+1−HbHo1+cosβ21+HbHsin3β2Hay model [10].
(20)Rd=HbHoRb+1−HbHo1+cosβ2The daily beam radiation incident on the horizontal surface is the difference between the global radiation on the horizontal surface and the diffuse radiation on the horizontal surface, as given in equation (21) [27].
(21)Hb=H−HdSkarveit and Olseth model [18].
(22)Rd=HbHoRb+Ωcosβ+1−HbHo−Ω1+cosβ2WhereΩ=Max0,0.3−2Hb/HoSteven and Unsworth model [19].
(23)Rd=0.51Rb+1+cosβ2−1.741.26πsinβ−βπ180cosβ−πsin2β2


#### 2.3.3. Reflected Radiation (HR)

The daily ground reflected radiation can be estimated using equation (24) [7].
(24)HR=Hρ1−cosβ2Whereρ is the albedo or ground reflectance. Its value varies from 0.2 for a snow-free surface and 0.6 with fresh snow [32]. In this study, it is taken to be 0.2 [12].Therefore, the total solar radiation on the tilted surface can be evaluated as [33]:
(25)HT=H−HdRb+Hρ1−cosβ2+HdRd


### 2.4. Statical Methods of Model Evaluation

To evaluate the accuracy of the proposed models’ several statistical evaluation indices are used. These include the root mean square error (RMSE), mean bias error (MBE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R2) which are defined by equation (26), respectively [34–37].
(26)RMSE=1n∑i=1nβcal,i−βpredt,i2MBE=1n∑i=1nβcal,i−βpredt,iMAE=1n∑i=1nβcal,i−βpredt,iMAPE=1n∑i=1nβcal,i−βpredt,iβcal,ix100%R2=1−∑i=1nβcal,i−βpredt,i2∑i=1nβcal,i2


### 2.5. Procedure for the Determination of the Optimal Tilt Angle

A MATLAB code was developed to obtain the optimal tilt angle for a solar module facing due south. The diffuse solar radiation and the monthly average daily total solar radiation on the inclined surface of all study sites were determined using both isotropic and anisotropic models. The angle varies from 0o to 90o with 1o step and the maximum solar radiation incident on the tilted surface is determined. The optimal tilt angles were determined by finding the angle for which the total solar radiation on the surface of the solar module was maximum.The optimal tilt angle generic models have been developed that estimate the seasonal and yearly optimal tilt angle at any location in Ethiopia using the data obtained from 32 study locations using R 4.1.2. statistical programming software. The capability of the developed models to estimate the seasonal and yearly optimal tilt angle for any location was tested using the data obtained from 12 study sites and their accuracy were validated using RMSE, MBE, MAE, MAPE, and R2 statistical indices. The summary of the workflow is given in Figure 2.Figure 2
Flow chart for the procedure in determining the optimal tilt angle model.


## 2.1. Data Set

To develop the generic optimal tilt angle models, 44 cities of Ethiopia are randomly selected and their monthly average daily global horizontal radiation was obtained from the National Aeronautical and Space Administration (NASA) database from 01/01/2010 to 31/12/2020 as shown in Figure1. A sample monthly average daily total horizontal solar radiation data for Bahir Dar city is shown in Table 2. The monthly average extraterrestrial radiation, clearness index, and diffuse radiations are evaluated for each study site using the solar radiation model and is discussed in the following subsection.Figure 1
Study sites.Table 2
Geographical coordinates and monthly average daily global horizontal radiation data of Bahir Dar city.
Latitude (o)Longitude (o)MonthGlobal horizontal radiation, H (kWh/m2/day)11.574237.3614January6.131February6.585March6.797April6.860May6.186June5.846July5.021August4.998September5.833October6.186November5.922December5.975


## 2.2. Solar Radiation on a Horizontal Surface

The solar radiation above the earth’s atmosphere is called extraterrestrial radiation. The daily extraterrestrial radiation on a horizontal surface can be calculated using equation (1) [7]
(1)Ho=24Gscπ1+0.033cos360n365cosϕcosδcosωs+πωs180sinϕsinδWhereωs is the sunset hour angle (degree) and computed as [7]:
(2)ωs=cos−1−tanϕtanδThe declination angle (δ) is the angle formed between the line joining the center of the earth and the sun and the equatorial plane. Its angle value changes over the year from 23.45o north to 23.45o south. The declination angle as a function of the day of the year (n), can be evaluated using the equation (3) [25]:
(3)δ=23.45osin360o284+n365Only a portion of extraterrestrial radiation reaches the earth’s horizontal surface, which is expressed using the clearness index (Kt). The monthly average daily clearness index is the ratio of the monthly average daily global radiation on a horizontal surface (H) to the monthly average daily global extraterrestrial radiation (Ho) and is evaluated as [11].
(4)Kt=HHoThe diffused radiation (Hd) on the horizontal surface is a function of the clearness index and global horizontal radiation and can be determined using equations (5) and (6) [27].Forωs≤81.4° and 0.3≤Kt≤0.8(5)Hd=H1.391−3.560Kt+4.189Kt2−2.137Kt3Forωs>81.4° and 0.3≤Kt≤0.8(6)Hd=H1.311−3.022Kt+3.427Kt2−1.821Kt3


## 2.3. Total Solar Radiation on a Tilted Surface

The monthly average daily total solar radiation on the inclined surface of the solar panel is composed of (1) beam or direct radiation, (2) diffuse radiation, and (3) reflected radiation [20]. Thus for the tilted surface, the incident total monthly average daily total radiation, HT (kWh/m2/day) is obtained using equation (7) [17].
(7)HT=HB+HD+HR


### 2.3.1. Beam Radiation (HB)

The total daily beam radiation on the tilted surface of the solar energy converter can be expressed as [12].
(8)HB=H−HdRbWhere Rb is the ratio between the monthly mean daily direct radiation on a horizontal surface to the inclined surface. Rb is defined as [7]:
(9)Rb=cosθcosθzThe cosine value of incident angle, cos (θ) can be defined as a combination of different solar angles as follows [28].
(10)cosθ=sinδsinϕcosβ−sinδcosϕsinβcosγ+cosδcosϕcosβcosω+cosδsinϕsinβcosγcosω+cosδsinβsinγsinωThe cosine of the zenith angle (cos (θz)) is computed by setting the panel tilt angle to zero (β =0), as [29].
(11)cosθz=cosϕcosδcosω+sinϕsinδHence, for the south-facing surface where the azimuth angle value is zero (γ =0o), the equation (7) can be written as [30]:
(12)Rb=cosϕ−βcosδsinωss+π/180ωsssinϕ−βsinδcosϕcosδsinωs+π/180ωssinϕsinδWhereωss is the sunset hour angle for the tilted surface for the mean day of the month, obtained by equation (13) [30].
(13)ωss=mincos−1−tanϕtanδcos−1−tanϕ−βtanδ


### 2.3.2. Diffuse Radiation (HD)

Different studies have shown that there are several empirical models to determine diffuse radiation. The sky diffuse radiation on the inclined surface of the panel can be determined by equation (14) [24]:
(14)HD=HdRdThe ratio of the average daily diffuse solar radiation on the tilted surface to the diffuse radiation on the horizontal surface, (Rd), can be evaluated by using different isotropic (equations (15)–(18) and anisotropic (equations ((19) (20) (22) and (23)) empirical formulas as followsIsotropic model: Isotropic models assume that the diffuse sky radiation is uniformly distributed over the skydome. Different researchers suggest different empirical formulas to calculate Rd, these are:Liu and Jordan’s model [31].
(15)Rd=1+cosβ2Badescu model [13].
(16)Rd=3+cos2β4Tian et al. model [14].
(17)Rd=1−β180Koronakis model [15].
(18)Rd=132+cosβAnisotropic model: The anisotropic model assumes that diffuse sky radiations are the sum of anisotropically distributed diffused radiation components in the circum-solar region (sky near the solar disc) and isotropically distributed diffuse components from the rest of the skydome.Reindl et al. model [22].
(19)Rd=HbHoRb+1−HbHo1+cosβ21+HbHsin3β2Hay model [10].
(20)Rd=HbHoRb+1−HbHo1+cosβ2The daily beam radiation incident on the horizontal surface is the difference between the global radiation on the horizontal surface and the diffuse radiation on the horizontal surface, as given in equation (21) [27].
(21)Hb=H−HdSkarveit and Olseth model [18].
(22)Rd=HbHoRb+Ωcosβ+1−HbHo−Ω1+cosβ2WhereΩ=Max0,0.3−2Hb/HoSteven and Unsworth model [19].
(23)Rd=0.51Rb+1+cosβ2−1.741.26πsinβ−βπ180cosβ−πsin2β2


### 2.3.3. Reflected Radiation (HR)

The daily ground reflected radiation can be estimated using equation (24) [7].
(24)HR=Hρ1−cosβ2Whereρ is the albedo or ground reflectance. Its value varies from 0.2 for a snow-free surface and 0.6 with fresh snow [32]. In this study, it is taken to be 0.2 [12].Therefore, the total solar radiation on the tilted surface can be evaluated as [33]:
(25)HT=H−HdRb+Hρ1−cosβ2+HdRd


## 2.3.1. Beam Radiation (HB)

The total daily beam radiation on the tilted surface of the solar energy converter can be expressed as [12].
(8)HB=H−HdRbWhere Rb is the ratio between the monthly mean daily direct radiation on a horizontal surface to the inclined surface. Rb is defined as [7]:
(9)Rb=cosθcosθzThe cosine value of incident angle, cos (θ) can be defined as a combination of different solar angles as follows [28].
(10)cosθ=sinδsinϕcosβ−sinδcosϕsinβcosγ+cosδcosϕcosβcosω+cosδsinϕsinβcosγcosω+cosδsinβsinγsinωThe cosine of the zenith angle (cos (θz)) is computed by setting the panel tilt angle to zero (β =0), as [29].
(11)cosθz=cosϕcosδcosω+sinϕsinδHence, for the south-facing surface where the azimuth angle value is zero (γ =0o), the equation (7) can be written as [30]:
(12)Rb=cosϕ−βcosδsinωss+π/180ωsssinϕ−βsinδcosϕcosδsinωs+π/180ωssinϕsinδWhereωss is the sunset hour angle for the tilted surface for the mean day of the month, obtained by equation (13) [30].
(13)ωss=mincos−1−tanϕtanδcos−1−tanϕ−βtanδ


## 2.3.2. Diffuse Radiation (HD)

Different studies have shown that there are several empirical models to determine diffuse radiation. The sky diffuse radiation on the inclined surface of the panel can be determined by equation (14) [24]:
(14)HD=HdRdThe ratio of the average daily diffuse solar radiation on the tilted surface to the diffuse radiation on the horizontal surface, (Rd), can be evaluated by using different isotropic (equations (15)–(18) and anisotropic (equations ((19) (20) (22) and (23)) empirical formulas as followsIsotropic model: Isotropic models assume that the diffuse sky radiation is uniformly distributed over the skydome. Different researchers suggest different empirical formulas to calculate Rd, these are:Liu and Jordan’s model [31].
(15)Rd=1+cosβ2Badescu model [13].
(16)Rd=3+cos2β4Tian et al. model [14].
(17)Rd=1−β180Koronakis model [15].
(18)Rd=132+cosβAnisotropic model: The anisotropic model assumes that diffuse sky radiations are the sum of anisotropically distributed diffused radiation components in the circum-solar region (sky near the solar disc) and isotropically distributed diffuse components from the rest of the skydome.Reindl et al. model [22].
(19)Rd=HbHoRb+1−HbHo1+cosβ21+HbHsin3β2Hay model [10].
(20)Rd=HbHoRb+1−HbHo1+cosβ2The daily beam radiation incident on the horizontal surface is the difference between the global radiation on the horizontal surface and the diffuse radiation on the horizontal surface, as given in equation (21) [27].
(21)Hb=H−HdSkarveit and Olseth model [18].
(22)Rd=HbHoRb+Ωcosβ+1−HbHo−Ω1+cosβ2WhereΩ=Max0,0.3−2Hb/HoSteven and Unsworth model [19].
(23)Rd=0.51Rb+1+cosβ2−1.741.26πsinβ−βπ180cosβ−πsin2β2


## 2.3.3. Reflected Radiation (HR)

The daily ground reflected radiation can be estimated using equation (24) [7].
(24)HR=Hρ1−cosβ2Whereρ is the albedo or ground reflectance. Its value varies from 0.2 for a snow-free surface and 0.6 with fresh snow [32]. In this study, it is taken to be 0.2 [12].Therefore, the total solar radiation on the tilted surface can be evaluated as [33]:
(25)HT=H−HdRb+Hρ1−cosβ2+HdRd


## 2.4. Statical Methods of Model Evaluation

To evaluate the accuracy of the proposed models’ several statistical evaluation indices are used. These include the root mean square error (RMSE), mean bias error (MBE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R2) which are defined by equation (26), respectively [34–37].
(26)RMSE=1n∑i=1nβcal,i−βpredt,i2MBE=1n∑i=1nβcal,i−βpredt,iMAE=1n∑i=1nβcal,i−βpredt,iMAPE=1n∑i=1nβcal,i−βpredt,iβcal,ix100%R2=1−∑i=1nβcal,i−βpredt,i2∑i=1nβcal,i2


## 2.5. Procedure for the Determination of the Optimal Tilt Angle

A MATLAB code was developed to obtain the optimal tilt angle for a solar module facing due south. The diffuse solar radiation and the monthly average daily total solar radiation on the inclined surface of all study sites were determined using both isotropic and anisotropic models. The angle varies from 0o to 90o with 1o step and the maximum solar radiation incident on the tilted surface is determined. The optimal tilt angles were determined by finding the angle for which the total solar radiation on the surface of the solar module was maximum.The optimal tilt angle generic models have been developed that estimate the seasonal and yearly optimal tilt angle at any location in Ethiopia using the data obtained from 32 study locations using R 4.1.2. statistical programming software. The capability of the developed models to estimate the seasonal and yearly optimal tilt angle for any location was tested using the data obtained from 12 study sites and their accuracy were validated using RMSE, MBE, MAE, MAPE, and R2 statistical indices. The summary of the workflow is given in Figure 2.Figure 2
Flow chart for the procedure in determining the optimal tilt angle model.


## 3. Results and Discussion

The determination of the optimal tilt angle is done for all study sites, and general seasonal and annual optimal tilt angle models have been developed that work for any location in Ethiopia. However, to minimize the paper size the detailed result is described for one study site, Bahir Dar city, that can be applied similarly to others too.


### 3.1. Daily Total Solar Radiation

The monthly average daily global, diffuse, and beam horizontal radiation and clearness index of Bahir Dar city are shown in Figure3. As indicated in the figure, the monthly average daily global horizontal solar radiation varies between 4.998 - 6.860kWh/m2/day, with the minimum and maximum clearness index of 0.477 and 0.707, respectively. The beam and diffuse components which account for 75.6% and 24.4%, respectively, of the global horizontal solar radiation are also depicted in the figure.Figure 3
Global horizontal radiation (H), beam radiation (Hb), diffuse radiation (Hd), and clearness index (Kt) of Bahir Dar city.The optimum monthly average daily total solar radiation on the surface of the module with the monthly optimal tilt angle was evaluated using 4 isotropic and 4 anisotropic models. The isotropic equation (17) model provides the minimum average total solar radiation at the optimum tilted surface and the maximum total solar radiation is obtained using an anisotropic model of equation (23). Anisotropic models of equations (20) and (22) provide the same result solar radiation and optimum tilt angle.The monthly optimum tilt angle using both isotropic and anisotropic models is 0o for April-August. For the other months, using the isotropic models, the optimal tilt angles are varied between 0o (in September) of equation (16) and 45o (in December) of equation (17). On the other hand, using anisotropic models, it varied between 8o (September) and 46o (December).Figures4 and 5 show the average daily total solar radiation and the optimum points on a south-facing inclined surface obtained using isotropic (equation (16)) and anisotropic (equation (19)) models with different tilt angle values. These models were selected since equations (16) and (19) preferred to estimate the total incident solar radiation on the inclined surface with the smallest statistical errors among all other models [10, 20].Figure 4
Monthly average daily solar radiation availability and an optimal value of a tilted surface using the isotropic model of equation (16), Bahir Dar city.Figure 5
Monthly average daily solar radiation availability and an optimal value of a tilted surface using an anisotropic model of equation (19), Bahir Dar.The monthly, seasonal, and yearly optimal tilt angles obtained by using isotropic and anisotropic models are shown in Figures6–8, respectively. As indicated in Figure 7, the seasonal optimal tilt angle using isotropic models were estimated at 35.33o (equation (16)) to 39.33o (equation (18)), 3.33o (equation (17)) to 5o (equation ((15) and (16)), 0o, and 19.67o (equation (17)) to 24.33o (equation (18)) for winter, spring, summer, and autumn, respectively. Using an anisotropic model, the seasonal optimal tilt angles were found to be between 40o to 41o, 5.33o to 6o, 0o, and 25o to 26.67o for winter, spring, summer, and autumn, respectively. The anisotropic model gives less variation in tilt angle value than isotropic models. For both models, the minimum is 0o in summer and the maximum is in winter which varies between 35.33o to 41o.Figure 6
Monthly optimal tilt angle evaluated by isotropic and anisotropic models for Bahir Dar city.Figure 7
Seasonal optimal tilt angle evaluated by isotropic and anisotropic models for Bahir Dar city.Figure 8
Yearly optimal tilt angle evaluated by isotropic and anisotropic model for Bahir Dar city.The yearly optimal tilt angles were evaluated and found to be between 14.92o to 17.17o using the isotropic models and 17.58o to 18.42o using the anisotropic model, given in Figure 8. The variation using the anisotropic model is very small as compared to the isotropic model. The detailed seasonal and annual optimal tilt angles of 32 and 12 study locations in Ethiopia used to develop and test the models are given in Tables 3 and 4, respectively.Table 3
The seasonal and yearly optimal tilt angle of 44 cities in Ethiopia using the isotropic model, equation (16).
CitiesWinterSpringSummerAutumnAnnualAdama32.333.33017.6713.33Addis Ababa333.33018.3313.67Adigrat374.6702316.17Arbaminch29.332.3301511.67Asela31.67301612.67Asosa33.333.67017.6713.67Assaita344020.6714.67Awash32.673.33017.3313.33Awassa31.33301612.58Axum37.67502316.42Bahir Dar35.334020.3314.92Butajira32.333017.3313.17Debre Birhan33.333.67018.6713.92Debre Markose33.673.6701914.08Degehbur32.673.33017.6713.42Dek Island364.3302115.33Dessie343.67019.6714.33Dilla302.6701511.92Diredawa33.333.67018.6713.92Dubti34.334020.6714.75Filtu292.33013.3311.17Finoteselam34.333.67019.3314.33Gambela29.673015.6712.08Gebreguracha33.333.67018.6713.92Geladi313016.3312.58Ginir313015.6712.42Godie302.6701511.92Gonder36.334.33021.3315.5Harar33.333.67018.3313.83Hosaena30.333016.3312.42Jima29.332.6701612Jinka282.3301411.08Kebri Dahir313015.6712.42Logia33.674020.6714.58Mega27.67201210.42Mekelle36.674.67022.3315.92Metekle34.33401914.33Metema36.674.67021.3315.67Mizan Teferi282.33014.6711.25Nekemt32.333.33017.6713.33Semera34.33402114.83Shashemene31.333016.3312.67Welyta Sodo30.332.67015.3312.08Ziway32.33301713.08Table 4
The seasonal and yearly optimal tilt angle of 44 cities in Ethiopia using the anisotropic model, equation (19).
CitiesWinterSpringSummerAutumnAnnualAdama384.33022.3316.17Addis Ababa38.334.33022.6716.33Adigrat436.3302819.33Arbaminch35.333.33019.3314.50Asela37.334021.3315.67Asosa39.334.6702316.75Assaita40.335.33025.3317.75Awash38.334.33022.3316.25Awassa36.333.67020.3315.08Axum436.3302819.33Bahir Dar40.675.33025.3317.83Butajira37.334021.6715.75Debre Birhan394.67023.3316.75Debre Markose39.33502417.08Degehbur37.674.33022.3316.08Dek Island415.33025.6718Dessie40.33502517.58Dilla35.673.33019.6714.67Diredawa394.67023.3316.75Dubti40.335.33025.6717.83Filtu34.673018.3314Finoteselam405024.3317.33Gambela374021.3315.58Gebreguracha39.334.67023.3316.83Geladi36.333.67020.6715.17Ginir36.333.67020.6715.17Godie36.333.67020.6715.17Gonder41.675.67026.3318.42Harar38.334.67023.3316.58Hosaena36.674021.3315.50Jima36.333.6702115.25Jinka34.673.33018.6714.17Kebri Dahir36.333.67020.3315.08Logia40.335.33025.6717.83Mega33.332.67017.3313.33Mekelle42.336027.3318.92Metekle405024.3317.33Metema426026.3318.58Mizan Teferi35.673.6702014.83Nekemt38.334.33022.3316.25Semera40.335.33025.6717.83Shashemene36.333.6702115.25Welyta Sodo36.333.67020.3315.08Ziway37.334021.3315.67The monthly average daily solar radiation at monthly, seasonal, and yearly optimal tilt angles are shown in Figure9. The monthly average total solar radiation resulting from the monthly optimal tilt angle is higher than the seasonal and the solar radiation with the fixed yearly optimal tilt angle is the lowest. As indicated in the figure, the variation in total solar radiation due to monthly and seasonal optimal tilt angles is very small. However, there is a significant change in total solar radiation when using a seasonal tilt angle than a yearly optimal tilt angle (fixed). In all cases, the estimated total solar radiation at the optimal tilt angle with the anisotropic equation (23) results in the highest value than other models.Figure 9
Monthly average daily solar radiation at monthly, seasonal, and yearly optimum tilt angle using the isotropic and anisotropic models, Bahir Dar city.


### 3.2. Losses in Solar Radiation

The percentage seasonal and yearly total solar radiation losses of an inclined surface compared to the solar radiation at the monthly optimal tilt angle were evaluated using equation (27) [16].
(27)Hlosses%=HTβ=βopt,M−HTβ=βopt,SYHTβ=βopt,Mx100The seasonal and yearly percentage total solar radiation losses on a tilted surface for Bahir Dar using isotropic and anisotropic models are shown in Figures10 and 11, respectively. The minimum and maximum seasonal losses obtained by the isotropic model are 0.872% and 2.30%, respectively. Using an anisotropic model, the losses are between 0.941% to 1.053%. The yearly optimal tilt angle has a loss of 5.11% to 6.275% using the isotropic model and 5.72% to 6.346% using the anisotropic model, which shows a significant loss when using a fixed annual optimal tilt angle than the optimal tilt angle that varied seasonally.Figure 10
Solar radiation seasonal losses, annual losses relative to monthly and annual losses relative to seasonal (%).Figure 11
Solar radiation seasonal losses, annual losses relative to monthly and annual losses relative to seasonal (%).The percentage gain of total solar radiation on a tilted surface compared to the horizontal surface is calculated by equation (28) [38]:
(28)Hgains%=HTβ=βopt,MSY−HTβ=0HTβ=0x100Using equation (28) the monthly, seasonal, and annual average total solar radiation gain of a tilted surface was evaluated for both isotropic and anisotropic models. For Bahir Dar city, the monthly average gain varies from 6.785% to 8.925% using isotropic and from 10.169% to 26.991% using the anisotropic model. The seasonal gains using the isotropic models vary between 5.723% to 7.532% and between 9.025% and 25.796% using the anisotropic model. Similarly, the yearly gain using isotropic and anisotropic models varied between 0.481% to 2.804% and 3.268% to 19.725%, respectively. From this result, the anisotropic model gives a better result than the isotropic model for monthly, seasonal, and yearly gains.


### 3.3. Developed Optimal Tilt Angle Models

In this study, the general models to predict the seasonal and annual optimal tilt angles were developed based on the linear regression model, shown in Table5. Of the total data, 73% of the data (32 cities) were used to obtain the model and 27% of the data (12 cities) were used to test/evaluate the model. These models accurately predict the seasonal and annual optimal tilt angle of the solar panel for any city by using the location’s latitude. The result from the developed model is in close agreement with the result obtained in [26]. The scatter plots with the regression developed models for winter, spring, autumn, and annual using equations (16) and (19) are given in Figures 12 and 13, respectively.Table 5
Developed isotropic and anisotropic seasonal and annual optimal tilt angle models.
TimeIsotropic modelEqu (15)Equ (16)Equ (17)Equ (18)Winterβopt=0.956ϕ+26.474βopt=1.047ϕ+22.7561βopt=1.1280ϕ+24.1350βopt=0.961ϕ+28.056Springβopt=0.340ϕ+0.911βopt=0.297ϕ+0.645βopt=0.4980ϕ−2.6410βopt=0.347ϕ+1.117Summerβopt=0βopt=0βopt=0βopt=0Autumnβopt=0.998ϕ+11.332βopt=1.004ϕ+8.636βopt=1.009ϕ+7.958βopt=0.996ϕ+12.674Annualβopt=0.585ϕ+9.566βopt=0.597ϕ+7.909βopt=0.657ϕ+7.439βopt=0.587ϕ+10.356Anisotropic modelEqu (19)Equ (20)Equ (22)Equ (23)Winterβopt=0.965ϕ+29.426βopt=0.977ϕ+28.528βopt=0.977ϕ+28.528βopt=0.876ϕ+30.818Springβopt=0.366ϕ+1.088βopt=0.370ϕ+0.978βopt=0.370ϕ+0.978βopt=0.363ϕ+1.708Summerβopt=0βopt=0βopt=0βopt=0Autumnβopt=1.030ϕ+13.282βopt=1.051ϕ+12.517βopt=1.051ϕ+12.517βopt=0.959ϕ+15.338Annualβopt=0.602ϕ+10.826βopt=0.614ϕ+10.363βopt=0.614ϕ+10.363βopt=0.562ϕ+11.842Figure 12
Regression model for seasonal and annual optimal tilt angle using Equation (16).
(a)(b)(c)(d)Figure 13
Regression model for seasonal and annual optimal tilt angle using Equation (19).
(a)(b)(c)(d)The optimal tilt angle of the study city is evaluated for the summer season using the isotropic and anisotropic models. For all isotropic and anisotropic models at this season, the optimal tilt angle is obtained to be zero. Its scatters plot of the developed regression model for both isotropic and anisotropic cases is shown in Figure14.Figure 14
Regression model for summer optimal tilt angle using Equations ((15)–(20)), (22), and (23).


#### 3.3.1. Validation of the Results

Based on the analysis of accuracy testing tools, the proposed models are evaluated and the results are shown in Tables6 and 7. As indicated in Tables 6 and 7, the smallest statistical error indicates that the developed model can accurately estimate both the seasonal and annual optimal tilt angle for any location in Ethiopia. The isotropic model equation (18) and from anisotropic model equations (19) and (23) have smaller errors, which is in agreement with the literature [10, 20]. The direction of the developed model to the actual value is determined by MBE. The anisotropic model of equation (23) has a negative MBE which indicates that the predicted values are overestimated than the actual values. The coefficient of determination (R2) indicates to what extent the predicted values are close to the actual. The minimum R2 is 0.8836 (88.36%) obtained for equation (17) and R2 of nearly 1 (100%) is obtained by an anisotropic model of equations (19) and (23), which indicates that the anisotropic model of equations (19) and (23) predicts more accurately than isotropic models.Table 6
Statistical errors of the developed equation using isotropic and anisotropic.
TimeIsotropic modelsRMSEMBEMAEMAPER2RMSEMBEMAEMAPER2Equ (15)Equ (16)Winter0.8550.1700.6572.000.99940.9260.7310.8592.770.9991Spring0.1070.0680.0732.070.99910.2050.1450.1736.040.9953Autumn0.3540.0680.3191.730.99960.4750.1610.4232.700.9991Annual0.2040.1720.1721.240.99980.3680.2900.3102.470.9991Equ (17)Equ (18)Winter0.8210.6530.7582.310.99940.3680.2790.3080.870.9999Spring0.5720.4730.4898.020.88360.1320.1230.1233.410.9988Autumn0.6280.1680.5173.560.99840.2520.0440.1940.940.9998Annual0.3700.2580.2982.350.99910.1700.1360.1541.050.9999Table 7
Performance evaluation of the anisotropic developed optimal tilt angle model using statistical indices.
TimeAnisotropic modelsRMSEMBEMAEMAPER2RMSEMBEMAEMAPER2Equ (19)Equ (20)Winter0.2590.1250.2130.581.0000.4590.3280.4051.120.9998Spring0.0920.0350.0731.960.99940.1170.0890.0962.840.9990Autumn0.2570.0820.2121.000.99980.2370.1360.1780.870.9999Annual0.1390.0940.1190.780.99990.2900.2560.2631.790.9996Equ (22)Equ (23)Winter0.4590.3280.4051.120.99980.094-0.0330.0800.211.000Spring0.1170.0890.0962.840.99900.109-0.0100.0862.240.9994Autumn0.2370.1360.1780.870.99990.203-0.0980.1710.800.9999Annual0.2220.1780.1981.340.99980.047-0.0030.0400.251.000


#### 3.3.2. The Developed Model in Comparison with Literature and Other Software

The results of this study were also compared with the optimal tilt angle obtained from the published works and online software. The annual optimal tilt angle (βopt) of this study location is determined from the correlation obtained from different literature. PVGIS and PVWatt are online software used to size PV systems at an optimized tilt angle. The annual optimal tilt angle comparison of the sample study site (Bahir Dar city) by using literature, online software, and this study is given in Table 8.Table 8
Annual optimal tilt angle of Bahir Dar city by this study, literature, and online software.
LiteratureModelβoptYunus Khan et al. [32]βopt=1.3793+ϕ1.2011+ϕ−0.014404+ϕ0.00008050913.48Duffie and Beckman [7]βopt=ϕ+15±1526.57Wessley et al. [39]βopt=0.823ϕ+8.827418.35Darhmaoui and Lahjouji [40]βopt=1.25351ϕ−0.00728944ϕ213.53Nicolas-Martín et al. [23]βopt=−0.007021ϕ2+1.091ϕ+2.13213.82PVGIS [41]Online software17PVWatt [42]Online software16This studyβopt=0.585ϕ+9.566 Equ (15)16.34βopt=0.597ϕ+7.909 Equ (16)14.82βopt=0.657ϕ+7.439 Equ (17)15.04βopt=0.587ϕ+10.356 Equ (18)17.15βopt=0.602ϕ+10.826 Equ (19)17.79βopt=0.614ϕ+10.363 Equ (20)17.47βopt=0.614ϕ+10.363 Equ (22)17.47βopt=0.562ϕ+11.842 Equ (23)18.35Yunus Khan et al. [33] proposed a latitude-based correlation of the optimal tilt angle model. The annual optimal tilt angle obtained from this literature is 13.48o which has a minor deviation of 1.34o to 4.87o compared to this study. A similar relation has been given in the literature by Darhmaoui and Lahjouji [40]. Wessley et al. [39] developed an optimal tilt angle correlation model to determine the Indian cities’ optimal tilt angle. A small optimal tilt angle deviation of 0o to 3.53o has been observed between this literature and the current study. The current study also compared with the model proposed in the literature by Nicolas-Martín et al. [23] and a maximum of 4.53o deviations of the optimal tilt angle has been observed. However, the optimal tilt angle model proposed by Duffie and Beckman [7] results in a minimum of 8.22o tilt angle deviation as compared to this study. This is due to the models proposed in all literature are not equally applicable for different locations.The results of this study were also validated by comparing with the online PV sizing software such as PVGIS and PVWatt. This software generates a very close optimal tilt angle to the current study and the observed minimum and maximum deviation with the study are 0.15 to 2.18 and 0.34 to 2.35 for PVGIS and PVWatt, respectively. These comparisons also validate the performance of the proposed system in determining the optimal tilt angle.Therefore, the result of this study gives an optimal tilt angle, and the developed model can be used in real applications of photovoltaic engineering to install the PV module at any location in Ethiopia. Installing the photovoltaic module with the suggested optimal tilt angle helps to obtain the maximum solar radiation on the surface of the module, hence gives the maximum energy output.


## 3.1. Daily Total Solar Radiation

The monthly average daily global, diffuse, and beam horizontal radiation and clearness index of Bahir Dar city are shown in Figure3. As indicated in the figure, the monthly average daily global horizontal solar radiation varies between 4.998 - 6.860kWh/m2/day, with the minimum and maximum clearness index of 0.477 and 0.707, respectively. The beam and diffuse components which account for 75.6% and 24.4%, respectively, of the global horizontal solar radiation are also depicted in the figure.Figure 3
Global horizontal radiation (H), beam radiation (Hb), diffuse radiation (Hd), and clearness index (Kt) of Bahir Dar city.The optimum monthly average daily total solar radiation on the surface of the module with the monthly optimal tilt angle was evaluated using 4 isotropic and 4 anisotropic models. The isotropic equation (17) model provides the minimum average total solar radiation at the optimum tilted surface and the maximum total solar radiation is obtained using an anisotropic model of equation (23). Anisotropic models of equations (20) and (22) provide the same result solar radiation and optimum tilt angle.The monthly optimum tilt angle using both isotropic and anisotropic models is 0o for April-August. For the other months, using the isotropic models, the optimal tilt angles are varied between 0o (in September) of equation (16) and 45o (in December) of equation (17). On the other hand, using anisotropic models, it varied between 8o (September) and 46o (December).Figures4 and 5 show the average daily total solar radiation and the optimum points on a south-facing inclined surface obtained using isotropic (equation (16)) and anisotropic (equation (19)) models with different tilt angle values. These models were selected since equations (16) and (19) preferred to estimate the total incident solar radiation on the inclined surface with the smallest statistical errors among all other models [10, 20].Figure 4
Monthly average daily solar radiation availability and an optimal value of a tilted surface using the isotropic model of equation (16), Bahir Dar city.Figure 5
Monthly average daily solar radiation availability and an optimal value of a tilted surface using an anisotropic model of equation (19), Bahir Dar.The monthly, seasonal, and yearly optimal tilt angles obtained by using isotropic and anisotropic models are shown in Figures6–8, respectively. As indicated in Figure 7, the seasonal optimal tilt angle using isotropic models were estimated at 35.33o (equation (16)) to 39.33o (equation (18)), 3.33o (equation (17)) to 5o (equation ((15) and (16)), 0o, and 19.67o (equation (17)) to 24.33o (equation (18)) for winter, spring, summer, and autumn, respectively. Using an anisotropic model, the seasonal optimal tilt angles were found to be between 40o to 41o, 5.33o to 6o, 0o, and 25o to 26.67o for winter, spring, summer, and autumn, respectively. The anisotropic model gives less variation in tilt angle value than isotropic models. For both models, the minimum is 0o in summer and the maximum is in winter which varies between 35.33o to 41o.Figure 6
Monthly optimal tilt angle evaluated by isotropic and anisotropic models for Bahir Dar city.Figure 7
Seasonal optimal tilt angle evaluated by isotropic and anisotropic models for Bahir Dar city.Figure 8
Yearly optimal tilt angle evaluated by isotropic and anisotropic model for Bahir Dar city.The yearly optimal tilt angles were evaluated and found to be between 14.92o to 17.17o using the isotropic models and 17.58o to 18.42o using the anisotropic model, given in Figure 8. The variation using the anisotropic model is very small as compared to the isotropic model. The detailed seasonal and annual optimal tilt angles of 32 and 12 study locations in Ethiopia used to develop and test the models are given in Tables 3 and 4, respectively.Table 3
The seasonal and yearly optimal tilt angle of 44 cities in Ethiopia using the isotropic model, equation (16).
CitiesWinterSpringSummerAutumnAnnualAdama32.333.33017.6713.33Addis Ababa333.33018.3313.67Adigrat374.6702316.17Arbaminch29.332.3301511.67Asela31.67301612.67Asosa33.333.67017.6713.67Assaita344020.6714.67Awash32.673.33017.3313.33Awassa31.33301612.58Axum37.67502316.42Bahir Dar35.334020.3314.92Butajira32.333017.3313.17Debre Birhan33.333.67018.6713.92Debre Markose33.673.6701914.08Degehbur32.673.33017.6713.42Dek Island364.3302115.33Dessie343.67019.6714.33Dilla302.6701511.92Diredawa33.333.67018.6713.92Dubti34.334020.6714.75Filtu292.33013.3311.17Finoteselam34.333.67019.3314.33Gambela29.673015.6712.08Gebreguracha33.333.67018.6713.92Geladi313016.3312.58Ginir313015.6712.42Godie302.6701511.92Gonder36.334.33021.3315.5Harar33.333.67018.3313.83Hosaena30.333016.3312.42Jima29.332.6701612Jinka282.3301411.08Kebri Dahir313015.6712.42Logia33.674020.6714.58Mega27.67201210.42Mekelle36.674.67022.3315.92Metekle34.33401914.33Metema36.674.67021.3315.67Mizan Teferi282.33014.6711.25Nekemt32.333.33017.6713.33Semera34.33402114.83Shashemene31.333016.3312.67Welyta Sodo30.332.67015.3312.08Ziway32.33301713.08Table 4
The seasonal and yearly optimal tilt angle of 44 cities in Ethiopia using the anisotropic model, equation (19).
CitiesWinterSpringSummerAutumnAnnualAdama384.33022.3316.17Addis Ababa38.334.33022.6716.33Adigrat436.3302819.33Arbaminch35.333.33019.3314.50Asela37.334021.3315.67Asosa39.334.6702316.75Assaita40.335.33025.3317.75Awash38.334.33022.3316.25Awassa36.333.67020.3315.08Axum436.3302819.33Bahir Dar40.675.33025.3317.83Butajira37.334021.6715.75Debre Birhan394.67023.3316.75Debre Markose39.33502417.08Degehbur37.674.33022.3316.08Dek Island415.33025.6718Dessie40.33502517.58Dilla35.673.33019.6714.67Diredawa394.67023.3316.75Dubti40.335.33025.6717.83Filtu34.673018.3314Finoteselam405024.3317.33Gambela374021.3315.58Gebreguracha39.334.67023.3316.83Geladi36.333.67020.6715.17Ginir36.333.67020.6715.17Godie36.333.67020.6715.17Gonder41.675.67026.3318.42Harar38.334.67023.3316.58Hosaena36.674021.3315.50Jima36.333.6702115.25Jinka34.673.33018.6714.17Kebri Dahir36.333.67020.3315.08Logia40.335.33025.6717.83Mega33.332.67017.3313.33Mekelle42.336027.3318.92Metekle405024.3317.33Metema426026.3318.58Mizan Teferi35.673.6702014.83Nekemt38.334.33022.3316.25Semera40.335.33025.6717.83Shashemene36.333.6702115.25Welyta Sodo36.333.67020.3315.08Ziway37.334021.3315.67The monthly average daily solar radiation at monthly, seasonal, and yearly optimal tilt angles are shown in Figure9. The monthly average total solar radiation resulting from the monthly optimal tilt angle is higher than the seasonal and the solar radiation with the fixed yearly optimal tilt angle is the lowest. As indicated in the figure, the variation in total solar radiation due to monthly and seasonal optimal tilt angles is very small. However, there is a significant change in total solar radiation when using a seasonal tilt angle than a yearly optimal tilt angle (fixed). In all cases, the estimated total solar radiation at the optimal tilt angle with the anisotropic equation (23) results in the highest value than other models.Figure 9
Monthly average daily solar radiation at monthly, seasonal, and yearly optimum tilt angle using the isotropic and anisotropic models, Bahir Dar city.


## 3.2. Losses in Solar Radiation

The percentage seasonal and yearly total solar radiation losses of an inclined surface compared to the solar radiation at the monthly optimal tilt angle were evaluated using equation (27) [16].
(27)Hlosses%=HTβ=βopt,M−HTβ=βopt,SYHTβ=βopt,Mx100The seasonal and yearly percentage total solar radiation losses on a tilted surface for Bahir Dar using isotropic and anisotropic models are shown in Figures10 and 11, respectively. The minimum and maximum seasonal losses obtained by the isotropic model are 0.872% and 2.30%, respectively. Using an anisotropic model, the losses are between 0.941% to 1.053%. The yearly optimal tilt angle has a loss of 5.11% to 6.275% using the isotropic model and 5.72% to 6.346% using the anisotropic model, which shows a significant loss when using a fixed annual optimal tilt angle than the optimal tilt angle that varied seasonally.Figure 10
Solar radiation seasonal losses, annual losses relative to monthly and annual losses relative to seasonal (%).Figure 11
Solar radiation seasonal losses, annual losses relative to monthly and annual losses relative to seasonal (%).The percentage gain of total solar radiation on a tilted surface compared to the horizontal surface is calculated by equation (28) [38]:
(28)Hgains%=HTβ=βopt,MSY−HTβ=0HTβ=0x100Using equation (28) the monthly, seasonal, and annual average total solar radiation gain of a tilted surface was evaluated for both isotropic and anisotropic models. For Bahir Dar city, the monthly average gain varies from 6.785% to 8.925% using isotropic and from 10.169% to 26.991% using the anisotropic model. The seasonal gains using the isotropic models vary between 5.723% to 7.532% and between 9.025% and 25.796% using the anisotropic model. Similarly, the yearly gain using isotropic and anisotropic models varied between 0.481% to 2.804% and 3.268% to 19.725%, respectively. From this result, the anisotropic model gives a better result than the isotropic model for monthly, seasonal, and yearly gains.


## 3.3. Developed Optimal Tilt Angle Models

In this study, the general models to predict the seasonal and annual optimal tilt angles were developed based on the linear regression model, shown in Table5. Of the total data, 73% of the data (32 cities) were used to obtain the model and 27% of the data (12 cities) were used to test/evaluate the model. These models accurately predict the seasonal and annual optimal tilt angle of the solar panel for any city by using the location’s latitude. The result from the developed model is in close agreement with the result obtained in [26]. The scatter plots with the regression developed models for winter, spring, autumn, and annual using equations (16) and (19) are given in Figures 12 and 13, respectively.Table 5
Developed isotropic and anisotropic seasonal and annual optimal tilt angle models.
TimeIsotropic modelEqu (15)Equ (16)Equ (17)Equ (18)Winterβopt=0.956ϕ+26.474βopt=1.047ϕ+22.7561βopt=1.1280ϕ+24.1350βopt=0.961ϕ+28.056Springβopt=0.340ϕ+0.911βopt=0.297ϕ+0.645βopt=0.4980ϕ−2.6410βopt=0.347ϕ+1.117Summerβopt=0βopt=0βopt=0βopt=0Autumnβopt=0.998ϕ+11.332βopt=1.004ϕ+8.636βopt=1.009ϕ+7.958βopt=0.996ϕ+12.674Annualβopt=0.585ϕ+9.566βopt=0.597ϕ+7.909βopt=0.657ϕ+7.439βopt=0.587ϕ+10.356Anisotropic modelEqu (19)Equ (20)Equ (22)Equ (23)Winterβopt=0.965ϕ+29.426βopt=0.977ϕ+28.528βopt=0.977ϕ+28.528βopt=0.876ϕ+30.818Springβopt=0.366ϕ+1.088βopt=0.370ϕ+0.978βopt=0.370ϕ+0.978βopt=0.363ϕ+1.708Summerβopt=0βopt=0βopt=0βopt=0Autumnβopt=1.030ϕ+13.282βopt=1.051ϕ+12.517βopt=1.051ϕ+12.517βopt=0.959ϕ+15.338Annualβopt=0.602ϕ+10.826βopt=0.614ϕ+10.363βopt=0.614ϕ+10.363βopt=0.562ϕ+11.842Figure 12
Regression model for seasonal and annual optimal tilt angle using Equation (16).
(a)(b)(c)(d)Figure 13
Regression model for seasonal and annual optimal tilt angle using Equation (19).
(a)(b)(c)(d)The optimal tilt angle of the study city is evaluated for the summer season using the isotropic and anisotropic models. For all isotropic and anisotropic models at this season, the optimal tilt angle is obtained to be zero. Its scatters plot of the developed regression model for both isotropic and anisotropic cases is shown in Figure14.Figure 14
Regression model for summer optimal tilt angle using Equations ((15)–(20)), (22), and (23).


### 3.3.1. Validation of the Results

Based on the analysis of accuracy testing tools, the proposed models are evaluated and the results are shown in Tables6 and 7. As indicated in Tables 6 and 7, the smallest statistical error indicates that the developed model can accurately estimate both the seasonal and annual optimal tilt angle for any location in Ethiopia. The isotropic model equation (18) and from anisotropic model equations (19) and (23) have smaller errors, which is in agreement with the literature [10, 20]. The direction of the developed model to the actual value is determined by MBE. The anisotropic model of equation (23) has a negative MBE which indicates that the predicted values are overestimated than the actual values. The coefficient of determination (R2) indicates to what extent the predicted values are close to the actual. The minimum R2 is 0.8836 (88.36%) obtained for equation (17) and R2 of nearly 1 (100%) is obtained by an anisotropic model of equations (19) and (23), which indicates that the anisotropic model of equations (19) and (23) predicts more accurately than isotropic models.Table 6
Statistical errors of the developed equation using isotropic and anisotropic.
TimeIsotropic modelsRMSEMBEMAEMAPER2RMSEMBEMAEMAPER2Equ (15)Equ (16)Winter0.8550.1700.6572.000.99940.9260.7310.8592.770.9991Spring0.1070.0680.0732.070.99910.2050.1450.1736.040.9953Autumn0.3540.0680.3191.730.99960.4750.1610.4232.700.9991Annual0.2040.1720.1721.240.99980.3680.2900.3102.470.9991Equ (17)Equ (18)Winter0.8210.6530.7582.310.99940.3680.2790.3080.870.9999Spring0.5720.4730.4898.020.88360.1320.1230.1233.410.9988Autumn0.6280.1680.5173.560.99840.2520.0440.1940.940.9998Annual0.3700.2580.2982.350.99910.1700.1360.1541.050.9999Table 7
Performance evaluation of the anisotropic developed optimal tilt angle model using statistical indices.
TimeAnisotropic modelsRMSEMBEMAEMAPER2RMSEMBEMAEMAPER2Equ (19)Equ (20)Winter0.2590.1250.2130.581.0000.4590.3280.4051.120.9998Spring0.0920.0350.0731.960.99940.1170.0890.0962.840.9990Autumn0.2570.0820.2121.000.99980.2370.1360.1780.870.9999Annual0.1390.0940.1190.780.99990.2900.2560.2631.790.9996Equ (22)Equ (23)Winter0.4590.3280.4051.120.99980.094-0.0330.0800.211.000Spring0.1170.0890.0962.840.99900.109-0.0100.0862.240.9994Autumn0.2370.1360.1780.870.99990.203-0.0980.1710.800.9999Annual0.2220.1780.1981.340.99980.047-0.0030.0400.251.000


### 3.3.2. The Developed Model in Comparison with Literature and Other Software

The results of this study were also compared with the optimal tilt angle obtained from the published works and online software. The annual optimal tilt angle (βopt) of this study location is determined from the correlation obtained from different literature. PVGIS and PVWatt are online software used to size PV systems at an optimized tilt angle. The annual optimal tilt angle comparison of the sample study site (Bahir Dar city) by using literature, online software, and this study is given in Table 8.Table 8
Annual optimal tilt angle of Bahir Dar city by this study, literature, and online software.
LiteratureModelβoptYunus Khan et al. [32]βopt=1.3793+ϕ1.2011+ϕ−0.014404+ϕ0.00008050913.48Duffie and Beckman [7]βopt=ϕ+15±1526.57Wessley et al. [39]βopt=0.823ϕ+8.827418.35Darhmaoui and Lahjouji [40]βopt=1.25351ϕ−0.00728944ϕ213.53Nicolas-Martín et al. [23]βopt=−0.007021ϕ2+1.091ϕ+2.13213.82PVGIS [41]Online software17PVWatt [42]Online software16This studyβopt=0.585ϕ+9.566 Equ (15)16.34βopt=0.597ϕ+7.909 Equ (16)14.82βopt=0.657ϕ+7.439 Equ (17)15.04βopt=0.587ϕ+10.356 Equ (18)17.15βopt=0.602ϕ+10.826 Equ (19)17.79βopt=0.614ϕ+10.363 Equ (20)17.47βopt=0.614ϕ+10.363 Equ (22)17.47βopt=0.562ϕ+11.842 Equ (23)18.35Yunus Khan et al. [33] proposed a latitude-based correlation of the optimal tilt angle model. The annual optimal tilt angle obtained from this literature is 13.48o which has a minor deviation of 1.34o to 4.87o compared to this study. A similar relation has been given in the literature by Darhmaoui and Lahjouji [40]. Wessley et al. [39] developed an optimal tilt angle correlation model to determine the Indian cities’ optimal tilt angle. A small optimal tilt angle deviation of 0o to 3.53o has been observed between this literature and the current study. The current study also compared with the model proposed in the literature by Nicolas-Martín et al. [23] and a maximum of 4.53o deviations of the optimal tilt angle has been observed. However, the optimal tilt angle model proposed by Duffie and Beckman [7] results in a minimum of 8.22o tilt angle deviation as compared to this study. This is due to the models proposed in all literature are not equally applicable for different locations.The results of this study were also validated by comparing with the online PV sizing software such as PVGIS and PVWatt. This software generates a very close optimal tilt angle to the current study and the observed minimum and maximum deviation with the study are 0.15 to 2.18 and 0.34 to 2.35 for PVGIS and PVWatt, respectively. These comparisons also validate the performance of the proposed system in determining the optimal tilt angle.Therefore, the result of this study gives an optimal tilt angle, and the developed model can be used in real applications of photovoltaic engineering to install the PV module at any location in Ethiopia. Installing the photovoltaic module with the suggested optimal tilt angle helps to obtain the maximum solar radiation on the surface of the module, hence gives the maximum energy output.


## 3.3.1. Validation of the Results

Based on the analysis of accuracy testing tools, the proposed models are evaluated and the results are shown in Tables6 and 7. As indicated in Tables 6 and 7, the smallest statistical error indicates that the developed model can accurately estimate both the seasonal and annual optimal tilt angle for any location in Ethiopia. The isotropic model equation (18) and from anisotropic model equations (19) and (23) have smaller errors, which is in agreement with the literature [10, 20]. The direction of the developed model to the actual value is determined by MBE. The anisotropic model of equation (23) has a negative MBE which indicates that the predicted values are overestimated than the actual values. The coefficient of determination (R2) indicates to what extent the predicted values are close to the actual. The minimum R2 is 0.8836 (88.36%) obtained for equation (17) and R2 of nearly 1 (100%) is obtained by an anisotropic model of equations (19) and (23), which indicates that the anisotropic model of equations (19) and (23) predicts more accurately than isotropic models.Table 6
Statistical errors of the developed equation using isotropic and anisotropic.
TimeIsotropic modelsRMSEMBEMAEMAPER2RMSEMBEMAEMAPER2Equ (15)Equ (16)Winter0.8550.1700.6572.000.99940.9260.7310.8592.770.9991Spring0.1070.0680.0732.070.99910.2050.1450.1736.040.9953Autumn0.3540.0680.3191.730.99960.4750.1610.4232.700.9991Annual0.2040.1720.1721.240.99980.3680.2900.3102.470.9991Equ (17)Equ (18)Winter0.8210.6530.7582.310.99940.3680.2790.3080.870.9999Spring0.5720.4730.4898.020.88360.1320.1230.1233.410.9988Autumn0.6280.1680.5173.560.99840.2520.0440.1940.940.9998Annual0.3700.2580.2982.350.99910.1700.1360.1541.050.9999Table 7
Performance evaluation of the anisotropic developed optimal tilt angle model using statistical indices.
TimeAnisotropic modelsRMSEMBEMAEMAPER2RMSEMBEMAEMAPER2Equ (19)Equ (20)Winter0.2590.1250.2130.581.0000.4590.3280.4051.120.9998Spring0.0920.0350.0731.960.99940.1170.0890.0962.840.9990Autumn0.2570.0820.2121.000.99980.2370.1360.1780.870.9999Annual0.1390.0940.1190.780.99990.2900.2560.2631.790.9996Equ (22)Equ (23)Winter0.4590.3280.4051.120.99980.094-0.0330.0800.211.000Spring0.1170.0890.0962.840.99900.109-0.0100.0862.240.9994Autumn0.2370.1360.1780.870.99990.203-0.0980.1710.800.9999Annual0.2220.1780.1981.340.99980.047-0.0030.0400.251.000


## 3.3.2. The Developed Model in Comparison with Literature and Other Software

The results of this study were also compared with the optimal tilt angle obtained from the published works and online software. The annual optimal tilt angle (βopt) of this study location is determined from the correlation obtained from different literature. PVGIS and PVWatt are online software used to size PV systems at an optimized tilt angle. The annual optimal tilt angle comparison of the sample study site (Bahir Dar city) by using literature, online software, and this study is given in Table 8.Table 8
Annual optimal tilt angle of Bahir Dar city by this study, literature, and online software.
LiteratureModelβoptYunus Khan et al. [32]βopt=1.3793+ϕ1.2011+ϕ−0.014404+ϕ0.00008050913.48Duffie and Beckman [7]βopt=ϕ+15±1526.57Wessley et al. [39]βopt=0.823ϕ+8.827418.35Darhmaoui and Lahjouji [40]βopt=1.25351ϕ−0.00728944ϕ213.53Nicolas-Martín et al. [23]βopt=−0.007021ϕ2+1.091ϕ+2.13213.82PVGIS [41]Online software17PVWatt [42]Online software16This studyβopt=0.585ϕ+9.566 Equ (15)16.34βopt=0.597ϕ+7.909 Equ (16)14.82βopt=0.657ϕ+7.439 Equ (17)15.04βopt=0.587ϕ+10.356 Equ (18)17.15βopt=0.602ϕ+10.826 Equ (19)17.79βopt=0.614ϕ+10.363 Equ (20)17.47βopt=0.614ϕ+10.363 Equ (22)17.47βopt=0.562ϕ+11.842 Equ (23)18.35Yunus Khan et al. [33] proposed a latitude-based correlation of the optimal tilt angle model. The annual optimal tilt angle obtained from this literature is 13.48o which has a minor deviation of 1.34o to 4.87o compared to this study. A similar relation has been given in the literature by Darhmaoui and Lahjouji [40]. Wessley et al. [39] developed an optimal tilt angle correlation model to determine the Indian cities’ optimal tilt angle. A small optimal tilt angle deviation of 0o to 3.53o has been observed between this literature and the current study. The current study also compared with the model proposed in the literature by Nicolas-Martín et al. [23] and a maximum of 4.53o deviations of the optimal tilt angle has been observed. However, the optimal tilt angle model proposed by Duffie and Beckman [7] results in a minimum of 8.22o tilt angle deviation as compared to this study. This is due to the models proposed in all literature are not equally applicable for different locations.The results of this study were also validated by comparing with the online PV sizing software such as PVGIS and PVWatt. This software generates a very close optimal tilt angle to the current study and the observed minimum and maximum deviation with the study are 0.15 to 2.18 and 0.34 to 2.35 for PVGIS and PVWatt, respectively. These comparisons also validate the performance of the proposed system in determining the optimal tilt angle.Therefore, the result of this study gives an optimal tilt angle, and the developed model can be used in real applications of photovoltaic engineering to install the PV module at any location in Ethiopia. Installing the photovoltaic module with the suggested optimal tilt angle helps to obtain the maximum solar radiation on the surface of the module, hence gives the maximum energy output.


## 4. Conclusion

In this study, the monthly, seasonal, and yearly optimal tilt angle of the solar module to harvest maximum incident solar radiation was determined for 44 cities of Ethiopia using isotropic and anisotropic models. Four isotropic and four anisotropic models were used to predict the diffuse solar radiation and the optimal tilt angle. The most important findings of this study are summarized as follows:(i)
Four isotropic and four anisotropic models were taken into consideration and their accuracy in predicting the optimal tilt angle is validated. The result revealed that the anisotropic models of Reindle et al., Steven and Unsworth predict the lower statistical errors(ii)
The yearly optimal tilt angle varied between 14.92o to 17.17o and 17.58o to 18.42o using isotropic and anisotropic models, respectively (as obtained for Bahir Dar). The anisotropic model gives a more consistent optimal tilt angle than the isotropic model(iii)
The seasonal varied optimal tilt angles are obtained between 35.33o to 39.33o in winter, 3.33o to 5o in spring, 0o in summer, and 19.67o to 24.33o in autumn using the isotropic model. The optimal tilt angle obtained using anisotropic models are 40o to 41o, 5.33o to 6o, 0o, and 25o to 26.67o for winter, spring, summer, and autumn, respectively. The minimum optimal tilt angle is 0o found in summer and the maximum is 35o to 41o, found in winter(iv)
Significant solar radiation gains are obtained when the solar module operates at a seasonal optimal tilt angle than a fixed (annual) optimal tilt angle. The seasonal gains of total solar radiation were 5.723% to 7.532% and between 9.025% to 25.796% using isotropic and anisotropic models, respectively. This shows that the anisotropic models have more gain as compared to the isotropic model(v)
A generic seasonal and annual optimal tilt angle model as a function of latitude was developed that estimates the optimal tilt angle accurately at any location in Ethiopia without using meteorological data. The accuracy of the developed model was evaluated using RMSE, MBE, MAE, MAPE, and R2. The model developed using anisotropically distributed diffused radiation results in a minim error, which is in agreement with the previous works in the literature. The model developed using Reindl et al. anisotropic model shows a smaller error of 0.139 (RMSE), 0.094 (MBE), 0.119 (MAE), and 0.78 (MAE) RMSE with a 0.9999 coefficient of determination (R2).The result of this study is verified using statistical indices, previous literature works, and online software. Online software has very close results with this work as those software use the local information to determine the optimal tilt angle. In this study, the developed models can determine the optimal tilt angle by taking only the latitude of the study site. However, this work can be further enhanced by considering other parameters in determining the optimal tilt angle models such as elevation and real-time measured solar radiation data.



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