A Novel Framework for Cost Optimization of Renewable Energy Installations: A Case Study of Nigeria

Abstract

The merits of utilizing renewable energy sources (RESs) in electricity generation, especially in the developing countries, are to improve the electricity access, economic development and energy sustainability. Nigeria is a developing country with an acute electricity problem. The country is blessed with rich renewable energy (RE) resources. However, most of these resources are yet to be exploited. A look at the energy sector in Nigeria suggests that for the country to be energy sufficient, it must embrace domestic RESs in its energy mix. However, RE technologies are capital intensive. Hence, by using Nigeria as a motivation, this study aims at developing a general framework that can be used for any country or region in determining the optimal total capacities of RESs to be installed in different locations, to increase the use of RE in a cost-effective manner. The designed optimization problem aims to minimize the total cost of installing RE technologies while satisfying some predetermined constraints that include demand and supply, RE potential, area and system reserve requirement. To this end, three different scenarios, namely prospective off-grid, on-grid, and all-off-grid are introduced. The first scenario aims at finding the optimal cost of installing RESs in order to improve electricity access at each off-grid installation location (a distribution company (DISCO) location with electricity access level below 50%). The second scenario optimizes cost of installing RESs for on-grid installation locations (DISCO locations (DLs) with electricity access level of 50% and above). The third scenario finds the optimal cost of installing off-grid RESs for all the DLs. Linear optimization technique is used to solve the problem. The results show that the total installation costs of the first and second scenarios (which means having off-grid installations for some DLs and having on-grid installations for other DLs) are $97.46 billion and $114.03 billion respectively, with a total cost of $211.49 billion. On the other hand, the result of third installation scenario (only off-grid installations for all DLs), is found to be $244.33 billion. These results reveal that the combination of off-grid and on-grid installations (first and second scenarios) has the minimum installation cost, for the case study of Nigeria. As the framework presented in this study is used to provide the minimum RE related total installation cost and related RE planning in Nigeria, it can also be useful for other countries or regions considering RE planning.

1. Introduction

The socio-economic development of a country is directly proportional to the amount of electricity utilized in that country [1]. However, according to Energy Access Outlook 2017 [2], about 1.1 billion people (14%) in the world have no access to electricity, out of which about 95% live in countries in developing Asia and sub-Saharan Africa. Hence, the development of most of these countries is mainly limited by their electricity production. Those countries should make investments in electricity infrastructure and power plants in order to increase their installed power capacity. Moreover, the substantial rise in the electricity demand, price of fossil fuels, the need to reduce air pollution, and improving energy security [3,4] necessitates commissioning more efficient and green power plants. However, both developed and developing countries are striving to improve their electricity generation through the use of renewable energy sources (RESs) [5]. In the last decades, many renewable energy (RE) technologies have become commercially available; such as wind, photovoltaic (PV), solar thermal, biomass, and various forms of hydro power [6]. Although the investment costs of RESs are capital intensive [4] the benefits associated with them are becoming more apparent due to the optimization models considering the cost [7]. If the cost problem is solved, RESs can be used to improve electricity access in developing countries where there is mismatch between the electricity supply and demand [8].

This study focuses on developing a general framework that can be used to determine the optimal cost of installing RE technologies in different locations in a country or a region, to increase RE utilization. In order to do that, novel methods that can be used to determine the best location for each RE technology, the best RE technology for each location, the available area for RESs installations and the electricity demand for a location were developed. The general framework developed in this study allows combining existing electricity facilities with increased use of RESs. It focuses on energy planning at the country or regional level, and it is illustrated by applying it to Nigeria. Nigeria is used as a case study not only due to the vast RE potential (which is defined as follows in this study; wind speed (m/s) for wind energy potential, solar radiation (kWh/m²/day) for solar energy etc.,) in the country, but also because of the current energy crisis bedeviling the country.

Despite the importance of electricity in an economy, Nigeria has not been able to generate adequate and reliable electricity to meet its demand [9]. As mentioned by Aliyu et al. [1], Nigeria is faced with acute electricity problem that has negatively affected the economic growth in the country. In 2016, about 40.7% of Nigerians had no access to grid electricity [10], and those with electricity access were faced with frequent power outages. Figure 1 shows that Nigeria’s electricity consumption per capita is below the average of sub-Saharan African countries [10]. It should be noted that Figure 1 is constructed by using the data available in [10]. As at March 2017, the country’s peak generation stood at 5074 MW against a peak demand of about 17,720 MW [11]. Mohammed et al. [12] revealed that about 63.82% of electricity generated in Nigeria comes from fossil fuels. However, Oyedepo [13] stated that the energy crisis afflicting the country will persist unless the government diversifies the energy sources and adopt new available technologies. In light of the above-mentioned circumstances, for Nigeria to achieve greater energy access, security and environmental sustainability, there is a need to incorporate more RE in its energy mix [14].

As shown in

Table 1. Nigeria’s renewable energy (RE) reserves [17,18].

Energy Source	Reserve
Animal Waste	61 million tons/yr
Crop Residue	83 million tons/yr
Large Hydro	11,235 MW
Small Hydro	3500 MW
Solar Radiation	3.5–7.5 kWh/m2/day
Wave and Tidal Energy	150,000 TJ/yr (16.6 × 106 toe/yr)
Wind	2–9 m/s at 10 m height

, Nigeria has great RE potential [15,16,17,18] which when harnessed properly can be helpful in solving the major electricity problems faced by the country. Accordingly, distributed variable RE technologies such as PV and wind systems have the potential of improving the country’s electricity access level, sustainability and security. Hence, the primary objective of the case study discussed in this paper is to improve electricity supply while optimizing the use of renewables in Nigeria.

This paper is organized as follows: Section 2 is devoted to the related literature review. Section 3 explains the methodology, and the main framework. Section 4 discusses the case study. Section 5 presents the results of the study. Section 6 gives the sensitivity analysis. Section 7 provides the discussion of the study. The conclusion of the study is presented in Section 8, along with future research directions.

2. Literature Review

2.1. Studies about Cost Optimization of Renewable Energy Systems

Several studies on cost optimization and planning of RESs have been conducted in literature. Ho et al. [19] developed a cost optimization model for RE-based distributed energy generation (DEG) system. Mixed Integer Linear Programming (MILP) method was used to determine the optimum cost of integrated solar and biomass system. The proposed model considered actual operation constraints due to availability of biomass, thermal power plant restriction and weather variation. The results elaborated that for optimum generation, biogas thermal power plant, direct fired biomass power plant and PV plant should generate 412 kW, 417 kW and 136 kW respectively. Similarly, Ferrer-Martí et al. [20] proposed a MILP model to solve the optimal cost of hybrid PV-wind plant in Peru. The mathematical model was used to design the micro grid that selects the best generation combination or option under certain constraints such as RE potential and load demand. The results show very significant cost reductions as the model can explore and allocate optimal design of micro grids. Milan et al. [21] focused on optimal sizing of renewable systems. Linear programming was used to find the overall system costs of three technology options that include PV, solar thermal collectors and heat pump. The study results revealed that heat pump combined with PV is the optimal configuration with the optimal cost of 75,200 euros.

Askarzadeh and Dos Santos [22] developed an optimal grid independent hybrid RESs in Kerman, Iran. Particle swarm optimization (PSO) was used to find the optimal values of the variables that include number of batteries, turbine swept area, and total area of PV panels. The results showed that wind/PV/battery system was the most cost-effective. Similarly, Lai et al. [23] developed an optimal sizing of stand-alone solar PV and storage system with anaerobic digestion biogas power plants in Kenya. PSO was used to determine the optimum size of PV and energy storage system (ESS) with biogas power plant. The proposed model considered the levelized cost of energy (LCOE) of the system while minimizing the energy imbalance between demand and generation due to intermittency of solar energy and anaerobic digestion constraints. The results showed that for optimum sizing, the PV plants, inverter, controller and vanadium redox flow battery should be 5 MW, 5000 kW, 5000 kW, and 5 MWh respectively. Also, Senjyu et al. [24] presented an optimal configuration of a power system with RE power production plants. The system comprised PV, battery, wind and diesel generators. The optimal configuration of the system was achieved by using genetic algorithm (GA). The simulation results showed that the total cost can be reduced by 10%.

Deshmukh and Deshmukh [25] developed a micro-level integrated RESs planning model for a rural area in India. In the study, multi-objective goal programming (MOGP) algorithm was used to optimally allocate RESs while considering some parameters such as load requirement, emissions, RE potential, social acceptance level and employment factor. The study revealed that biomass energy should be promoted for electrical energy generation, while solar thermal and biogas should be supported for cooking purposes. Similarly, Deshmukh and Deshmukh [26] proposed a MOGP approach to optimally allocate RESs that can meet the energy needs of three districts in India (Panthadiya, Bisanpura, and Morva). The study considered the optimal energy resource allocation for various end-uses. The simulation results revealed biomass, biogas and solar thermal as the preferred sources that can meet the load requirement of the Panthadiya, Bisanpura, and Morva. Niknam et al. [27] presented a Modified Honey Bee (MOP) algorithm for sizing and siting renewable generators. A modified honey bee mating algorithm was used to solve the optimization problem that consists of cost minimization, losses, voltage profile and emissions of the distributed system. The results show that proper sizing and siting of renewable electricity generations (REGs) are vital for emission reduction, voltage profile and cost reduction associated with the distribution system.

Chang [28] proposed an optimal design of hybrid RESs using Monte Carlo optimization technique. The model considered not only the equipment installation, including wind, PV, storage system and diesel generator, but also transmission and power allocation within the hybrid RESs in order to achieve minimum equipment cost. Similarly, Sharafi et al. [29] developed an optimal design of hybrid RESs in a building, with low to high RE ratio. Meta heuristic approach was used to model the potential use of PV panels, heat pump, wind turbine, biomass boiler, heat storage tank and solar heat collectors to produce RE for the building. The study showed that installing 73 kW wind turbine and 200 kW biomass boiler is the optimal option with a net present cost of C$705,180.

Iniyan and Sumathy [30] proposed an Optimal RE Model (OREM) that allocates different RESs for several end users. The energy demand, acceptance level, RE potential and reliability are used as constraints in the model. The results indicated that PV system could be used to cover the demand for pumping, cooling, lighting and heating to an extent of 16%, 12%, 6%, and 2% respectively. According to the authors, the bioenergy systems could generate 1%, 17%, 9%, 17%, and 14% of the total energy needed for cooking, heating, lighting, pumping and transportation respectively. Furthermore, 4% of the total energy required by pumping could be generated by using wind energy. Li et al. [31] developed a dynamic sizing and modeling optimization of stand-alone PV systems. Three different stand-alone PV systems that include PV/battery hybrid system, PV/fuel cell hybrid system and PV/battery/fuel cell hybrid system were evaluated. The simulation results showed that the optimal configuration was PV/battery/fuel cell hybrid system.

Jeppesen et al. [32] studied least cost and utility scale abatement from Australia’s national electricity market (NEM). The planning model formulated has many constraints which are vital when considering greenhouse gas abatement and widespread uptake of intermittent RE generation. The least cost pathways were evaluated by numerical optimization of utility scale generation, transmission, distribution and storage. The authors revealed that the estimated wholesale market cost of energy was found to be $ 37 per MWh. Rozali, et al. [33] developed a novel method for the design of a cost-effective hybrid power systems (HPS). The study incorporates power pinch analysis with a systematic hierarchical approach for resilient process screening for the design of HPS. The results showed that HPS design with minimum electricity targets will have a simple payback period of 10 years. Also, Razak et al. [34] developed an optimal design of RE hybrid system using HOMER simulation tool. The proposed model considered solar, hydro, wind, and storage system. The results revealed that if the demand load is increased to maximum capacity, the cost of energy can be reduced to about 50%. Also, the authors show that the net present cost, levelized cost of energy and total production was found to be RM 63,887, RM 1.019 kWh, and RM 8405 kWh respectively.

Yang and Nehorai [35] proposed an optimal design of cost minimization of RES, diesel generator and storage system using alternating direction method of. The authors formulated an optimization problem with the objective of minimizing the investment cost; maintenance cost and operational cost of RESs, storage system and diesel generators. The hourly load data from 2008 to 2012 used in generating the load data was obtained from Electricity reliability council of Texas while the renewable energy data used in the study was obtained from national solar radiation database. The results showed that to generate of 0.3432 MWh of energy, the optimum investment cost, operation and maintenance cost are 0.2196 and 0.0088 million dollars respectively.

Khan et al. [36] reviewed the optimal sizing techniques and cost analysis methodologies. The study focused on hybrid PV and wind energy systems. The authors shed light on various sizing strategies, various parameters of economic viability with logical advancements to improve their utilization, and future prospects of RESs. Also, the study reviewed the developments in optimization methods, cost analysis methods and reliability index for hybrid RE systems.

Kanase-Patil et al. [37] analyzed RESs integrated for off-grid rural electrification of remote area in India by using LINGO optimization software. The optimal system cost, system reliability and cost of energy were evaluated for four different RE technologies that include biomass, micro hydro, wind and solar. The results revealed the optimum cost and the cost of energy as Rs 19.44 lacs and Rs 3.36/kWh respectively. Arnette and Zobel [38] presented an optimization problem for RE development. In the paper, multi objective linear programming method was used to determine the optimal mix of RESs and the existing conventional facilities on a regional basis. The results revealed that the optimal cost and emissions were $6.25 billion and 166.6 million tons respectively.

2.2. Studies about Renewable Energy Potential in Nigeria

There are some previously reported studies on RE potential in Nigeria, for example, Mohammed et al. [12] reviewed the RE resources for distributed power generation in Nigeria and concluded that the country has vast RE resources with a potential of generating 697.15 TJ from crop residue and 455.80 PJ from animal waste in Lagos city alone. Ohunakin et al. [39] studied the wind energy evaluation for electricity generation using wind energy conversion systems (WECS) in seven selected locations in Nigeria. Weibull probability distribution was used to statistically examine the wind speed and power density in Yelwa, Sokoto, Gusau, Kaduna, Katsina, Zaria, and Kano. The study revealed that the average wind speeds at 10 m height for Yelwa, Sokoto, Gusau, Kaduna, Katsina, Zaria, and Kano are 3.61, 7.61, 6.09, 5.27, 7.45, 6.08, and 7.77 m/s respectively. Also, the yearly average power density was found to be 43.77, 320.09, 178.48, 109.30, 339.85, 169.27, and 368.92 W/m² in Yelwa, Sokoto, Gusau, Kaduna, Katsina, Zaria, and Kano respectively. Similarly, Ohunakin, and Akinnawonu [40] investigated the wind energy potential and economics of wind power generation in Jos, Plateau state, Nigeria, using 37 years (1971–2007) wind speed data at 10 m height. The analysis indicated that Jos has 8.6 m/s, 458 W/m² and 4013 kWh/year average wind speed, mean power density and energy production capacity respectively. Ajayi [41] reviewed the prospects and challenges of wind energy utilization in Nigeria. The study focused on the potential of generating electricity from wind in the northern states, mountainous regions, and some locations in the central and south-eastern states. The study revealed that despite vast wind energy potential across the states, there is no wind energy generating plant in the country.

Ohunakin et al. [42] studied the development of small hydro power (SHP) in Nigeria. The study examined the current development of hydropower in Nigeria with respect to the established energy sector reform act 2005. The authors revealed that from 1971 to 2005, hydropower sector has witnessed about 360% growth. Yet, SHP contributed only 5%. Similarly, Fagbohun [43] investigated the potential of SHP in Itapaji dam in Ekiti, Nigeria. The study analyzed the discharge rate, effective head and other parameters that determine the power that can be generated from the site. The results revealed that the dam has a maximum yearly discharge rate of 23.24 m³/s with a mean nominal flow discharge rate of 8.33 m³/s, and mean minimal flow of 1.78 m³/s. Also, the author showed that the dam has a potential of generating 1.30 MW of electricity.

Ohunakin et al. [44] studied the solar energy development and applications in Nigeria. The study focused on current solar energy utilization, drivers for RE development and barriers to large scale deployment. The authors elaborated that currently, there are about 61 solar projects across the country with most of them being in the northern region. Also, the study showed that emission reductions, power sector reform act, electricity demand, electricity access level and energy security are the major RE drivers in the country. Furthermore, some barriers such as grid unreliability, high cost of investment, incentives and government policies affect the development of solar energy in the country. Abur and Duvuna [45] studied the viability of solar energy in Benue state, Nigeria. The study revealed that the state receives its minimum and maximum solar radiation in the month of August and November with radiation of 14.57 MJ/m²/day and 20.16 MJ/m²/day respectively, and a yearly average solar radiation of 16.6 MJ/(m²·day) to 18.07 MJ/(m²·day).

It is vital to note that the above-mentioned approaches have been developed without novel methods on identifying the best technology for a location, the best location for a technology, available area for RESs installations, and electricity demand calculation. When necessary, these decisions/parameters have tended to contain simple estimates related to RESs potential, and available area for RESs installations, and, most often these data were derived from sources outside the provided research. This consists of a drawback to many optimization approaches which this study aims at solving. Unlike other studies, the objective of this study is to provide a general framework that can be used in any country or region. The framework is designed in a way that it considers the existing installed capacity and focuses on minimization of the total cost of installing RE technologies in different locations, to supply the gap between demand and supply, by using the best combination of various RE technologies. Thus, to the authors’ knowledge, there is no study in the literature that focus on developing a general cost optimization framework that can be used in deciding optimum amount of RE capacities to be installed in different locations in a region or country by considering the country’s peak power demand, and hence, the application of such a framework to Nigeria.

3. Methodology

This section is devoted to the problem setup and related descriptions. The primary goal is to develop a general framework that will be used to determine the optimal total capacities of RESs (solar, wind, hydro, biomass, geothermal, wave, etc.) to be installed in different locations in any region or country, in order to increase the amount of RE installations so that the power supply could be comparable to the peak power demand in a region/country. This is done by designing an optimization problem that minimizes the total cost of installing RESs that will be used to support the existing installed power capacity. Constraints of the designed optimization problem include; demand and supply, RE potential, landmass and system reserve requirement. Section 3.1 presents the novel methods developed to be used in optimization while Section 3.2 provides the optimization problem formulation, which includes objective function and constraints. The optimization problem formulation in this study is divided into off-grid and on-grid installation setups. The off-grid installation setup is mostly used when the region of analysis has low electricity access level. Conversely, the on-grid installation setup is mostly used when the region of analysis has high electricity access level.

3.1. Best Technology, Best Location, Available Area, and Electricity Demand Determination Methods

This sub-section presents some novel methods developed for this study, in order to determine the best location for a RE technology, best RE technology for a location, power demand of a location and, required area for RE installations.

3.1.1. Best Location and Best Technology Determination Method

In order to determine the best location and technology for RESs installations, Equations (1)–(6) are formulated. Thus, the comparative RE potential of technology j in location $i$, ${\overline{\mathrm{P}}}_{\mathrm{ij}}$, has been formed, as given in Equation (1).

$${\overline{\mathrm{P}}}_{\mathrm{ij}}=\frac{{\mathrm{P}}_{\mathrm{ij}}}{\frac{1}{\mathrm{n}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{ij}}}$$

(1)

where ${\mathrm{P}}_{\mathrm{ij}}$ is the RE potential (in terms of the amount of solar radiation (for solar resource potential), or the wind speed (for wind resource potential), etc.) of technology j in location $i$. It should be noted that, i = 1…n, and $\mathrm{j}$ = 1……$\mathrm{m}$ where $\mathrm{n}$ is the number of locations, m is the number of technologies and $\frac{1}{\mathrm{n}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{P}}_{\mathrm{ij}}$ is the average RE potential of technology $\mathrm{j}$ over all locations. Hence, Equation (1) is used to determine the comparative (normalized) potential of each technology in a given location. This equation is constructed in a way that it would allow comparison of different energy resources. As ${\overline{\mathrm{P}}}_{\mathrm{ij}}$ is unit less, it can be used to compare the potentials of different energy resources in each location, and similarly it can be used to determine the best location for each technology (mainly the location having the highest amount of comparative potential). In order to determine the best/worst technology/resource for each location, and best/worst location for each technology/resource, there is a need to form a special matrix. Thus, ${\mathrm{P}}_{\mathrm{N}}$ matrix, shown in Equation (2), is formed.

$${\mathrm{P}}_{\mathrm{N}}=\left[\begin{array}{cccc}{\overline{\mathrm{P}}}_{11}& {\overline{\mathrm{P}}}_{12}& \dots & {\overline{\mathrm{P}}}_{1\mathrm{m}}\\ {\overline{\mathrm{P}}}_{21}& {\overline{\mathrm{P}}}_{22}& \dots & {\overline{\mathrm{P}}}_{2\mathrm{m}}\\ {\overline{\mathrm{P}}}_{31}& {\overline{\mathrm{P}}}_{32}& \cdots & {\overline{\mathrm{P}}}_{3\mathrm{m}}\\ ⋮& ⋮& \cdots & ⋮\\ {\overline{\mathrm{P}}}_{\mathrm{n}1}& {\overline{\mathrm{P}}}_{\mathrm{n}2}& \cdots & {\overline{\mathrm{P}}}_{\mathrm{nm}}\end{array}\right]$$

(2)

${\mathrm{P}}_{\mathrm{N}}$ is the comparative potential matrix, showing comparative potentials of all technologies in all locations. ${\mathrm{P}}_{\mathrm{N}}$ is used to determine which locations should be favored for each technology and which technologies should be favored for each location. (Example from Equation (2); the minimum of the comparative RE potentials listed in row $i$, leads to the technology with minimum potential (worst technology) in location $i$. Similarly, the minimum of the comparative RE potentials listed in column $j$, leads to the worst location for a given technology) as presented in Equations (3) and (5) respectively. By using this matrix, the following set of variables can be defined.

$${\mathrm{WT}}_{\mathrm{i}}=\begin{array}{c}\begin{array}{c}\min \\ \mathrm{j}\end{array} \left\{{\overline{\mathrm{P}}}_{\mathrm{ij}}\right\}\end{array}\hspace{1em}\hspace{1em}\forall \mathrm{i}$$

(3)

$${\mathrm{BT}}_{\mathrm{i}}=\begin{array}{c}\begin{array}{c}\max \\ \mathrm{j}\end{array} \left\{{\overline{\mathrm{P}}}_{\mathrm{ij}}\right\}\end{array}\hspace{1em}\hspace{1em}\forall \mathrm{i}$$

(4)

$${\mathrm{WL}}_{\mathrm{j}}=\begin{array}{c}\begin{array}{c}\min \\ \mathrm{i}\end{array} \left\{{\overline{\mathrm{P}}}_{\mathrm{ij}}\right\}\end{array}\hspace{1em}\hspace{1em}\forall \mathrm{j}$$

(5)

$${\mathrm{BL}}_{\mathrm{j}}=\begin{array}{c}\begin{array}{c}\max \\ \mathrm{i}\end{array} \left\{{\overline{\mathrm{P}}}_{\mathrm{ij}}\right\}\end{array}\hspace{1em}\hspace{1em}\forall \mathrm{j}$$

(6)

where ${\mathrm{WT}}_{\mathrm{i}}$ and ${\mathrm{BT}}_{\mathrm{i}}$ are used to determine the worst and best technologies (in terms of potential) in location i, respectively. Similarly, ${\mathrm{WL}}_{\mathrm{j}}$ and ${\mathrm{BL}}_{\mathrm{j}}$ are used to determine the worst and best locations for technology j. From Equation (2), the best and worst locations/technologies can be determined using Equations (3)–(6).

3.1.2. Power Demand Calculation Method

Equation (7) can be used to calculate the total estimated power demand for the region of analysis.

$$\overline{{\mathrm{PD}}_{\mathrm{e}}}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\overline{\mathrm{ED}}}_{\mathrm{i}}$$

(7)

where ${\overline{\mathrm{ED}}}_{\mathrm{i}}$ is the scaled estimated power demand for location i in $\mathrm{MW}$ as shown in Equation (8). In order to clarify the usage of scaling factor and the power demand estimation procedure, the following lines are needed. First, the power supply for each location i, ${\mathrm{PS}}_{\mathrm{i}}$ should be obtained. After that, the peak demand for location i should be calculated (estimated) by applying a simple interpolation; if the supply of location i is ${\mathrm{PS}}_{\mathrm{i}}$, the demand (the required supply if the access level of that location was 100%), would be ${\mathrm{ED}}_{\mathrm{i}}$. By using ${\mathrm{ED}}_{\mathrm{i}}$ of each location, the total peak demand based on the electricity access levels (PD-EA) can be calculated, ${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{ED}}_{\mathrm{i}}$. However, the country’s reported peak demand, ${\mathrm{PD}}_{\mathrm{T}}$, can be different than the total PD-EA (${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{ED}}_{\mathrm{i}}$). In such cases, a scaling factor is needed in order to scale the PD-EAs so that ${\overline{\mathrm{ED}}}_{\mathrm{i}}$ could be obtained and the total estimated power ($\overline{{\mathrm{PD}}_{\mathrm{e}}}$) would match with ${\mathrm{PD}}_{\mathrm{T}}$, or provide a very close approximation to it. However, it should be noted that scaling factor is expected to change for each country or region. The related equations of the concepts mentioned above, are provided below:

$${\overline{\mathrm{ED}}}_{\mathrm{i}}=\mathrm{F}\times {\mathrm{ED}}_{\mathrm{i}}$$

(8)

where $\mathrm{F}$ is a scaling factor (the ratio of the total power demand to total estimated power demand of the region of analysis), and it is calculated as follows:

$$\mathrm{F}=\frac{{\mathrm{PD}}_{\mathrm{T}}}{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{ED}}_{\mathrm{i}}}$$

(9)

where ${\mathrm{PD}}_{\mathrm{T}}$ represents the total power demand of the region of analysis. ${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{ED}}_{\mathrm{i}}$ is the estimated total demand for the region of analysis where ${\mathrm{ED}}_{\mathrm{i}}$ is the estimated demand for location i when the access level is at 100%, and it is calculated as follows:

$${\mathrm{ED}}_{\mathrm{i}} = \frac{100 \times { \mathrm{PS}}_{\mathrm{i}} }{{\mathrm{a}}_{\mathrm{i}}}$$

(10)

where ${\mathrm{PS}}_{\mathrm{i}}$ is the power supply to each location $i$, and ${\mathrm{a}}_{\mathrm{i}}$ represents the electricity access level 0–100% in location $i$.

3.1.3. Required Area Calculation Method for RE Installations

The area required to install a specific RE technology can be calculated by using Equation (11). Thus, the available area in location $i$ that can be covered by technology $j$, ${\mathrm{AR}}_{\mathrm{i}}$, can be calculated as:

$${\mathrm{AR}}_{\mathrm{i}}=\left[\frac{1}{\mathrm{k}}{\displaystyle \sum }_{\mathrm{r}=1}^{\mathrm{k}}\frac{{{\displaystyle \sum }}_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{I}}_{\mathrm{jr}}{a}_j}{{\mathrm{A}}_{\mathrm{r}}}\right]\times {\mathrm{A}}_{\mathrm{i}}$$

(11)

where ${\mathrm{I}}_{\mathrm{jr}}$ is the total installed capacity of RE technology $j$(used in the analysis) for leading country r in terms of RE installation (LCRE) in $\mathrm{MW}$, ${\mathrm{a}}_{\mathrm{j}}$ is the per unit area covered by RE technology $j$ in $\frac{{\mathrm{km}}^2}{\mathrm{MW}}$. ${\mathrm{A}}_{\mathrm{r}}$ is the total area of leading country $r$ in ${\mathrm{km}}^2$. ${\mathrm{A}}_{\mathrm{i}}$ is the total area available in location i in ${\mathrm{km}}^2$. m is the total number of technologies, and $\mathrm{k}$ is the number of leading countries with RE installation (used in the analysis), used for Equation (11). It should be noted that the leading countries in terms of RE installations used in this study are Brazil, China, Germany, and USA [46]. The reason of selecting these countries is mainly the fact that they were the leading countries in terms of installed RE, by the time the data was collected. If the average percentage of landmass covered by RE in any leading country could be found, it would be useful for estimating the landmass required for another country, in order to make RE installations, as see in Equation (11).

3.2. Problem Formulation

3.2.1. Objective Function

The main objective of this study is to find the minimum cost of installing RE technologies in various locations while satisfying pre-determined constraints. The problem is mathematically formulated as a linear optimization (minimization) problem with the objective function shown in Equation (12).

$${\mathrm{IC}}_{\mathrm{T}}={\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{C}}_{\mathrm{j}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{ij}}$$

(12)

where ${\mathrm{IC}}_{\mathrm{T}}$ is the total installation cost in $, ${\mathrm{C}}_{\mathrm{j}}$ is the cost of installing $1\mathrm{MW}$ of technology j in $\$/\mathrm{MW}$, ${\mathrm{X}}_{\mathrm{ij}}$ represents the RE capacity for technology j to be installed in location i (prospective installed capacity), where i = 1…n and j = 1…m. In this framework ${{\mathrm{X}}_{\mathrm{ij}}}^{\prime }\mathrm{s}$ are the decision variables.

3.2.2. Constraints

Comparative Constraints: Zonal RE potentials can be determined by investigating RESs related data (wind speed, solar radiation etc.). Upon retrieving the related data, the set of sub-constraints in Equations (13)–(16), called comparative constraints, can be formed for each location i and for each technology $j$.

$${\mathrm{X}}_{\mathrm{ij}}\ge { \mathrm{X}}_{{\mathrm{ij}}_{\mathrm{WT}}}{ \mathrm{for} \mathrm{WT}}_{\mathrm{i}}\le {\overline{\mathrm{P}}}_{\mathrm{ij}}\hspace{1em}\forall \mathrm{i}$$

(13)

where $\mathrm{j}\ne {\mathrm{j}}_{\mathrm{WT}}$

$${\mathrm{X}}_{\mathrm{ij}} \le { \mathrm{X}}_{{\mathrm{ij}}_{\mathrm{BT}}}{ \mathrm{for} \mathrm{BT}}_{\mathrm{i}} \ge {\overline{\mathrm{P}}}_{\mathrm{ij}}\hspace{1em}\hspace{1em}\forall \mathrm{i}$$

(14)

where $\mathrm{j}\ne {\mathrm{j}}_{\mathrm{BT}}$

$${\mathrm{X}}_{\mathrm{ij}} \ge { \mathrm{X}}_{{\mathrm{i}}_{\mathrm{WL}}\mathrm{j}}{ \mathrm{for} \mathrm{WL}}_{\mathrm{j}} \le {\overline{\mathrm{P}}}_{\mathrm{ij}}\hspace{1em}\hspace{1em}\forall \mathrm{j}$$

(15)

where $\mathrm{i}\ne {\mathrm{i}}_{\mathrm{WL}}$

$${ \mathrm{X}}_{\mathrm{ij}} \le { \mathrm{X}}_{{\mathrm{i}}_{\mathrm{BL}}\mathrm{j}}{ \mathrm{for} \mathrm{BL}}_{\mathrm{j}} \ge {\overline{\mathrm{P}}}_{\mathrm{ij}}\hspace{1em}\hspace{1em}\forall \mathrm{j}$$

(16)

where $\mathrm{i}\ne {\mathrm{i}}_{\mathrm{BL}}$. ${\mathrm{X}}_{{\mathrm{ij}}_{\mathrm{WT}}}$ and ${\mathrm{X}}_{{\mathrm{ij}}_{\mathrm{BT}}}$ represent the prospective installed capacities of the worst and best renewable technologies in location i respectively. ${\mathrm{X}}_{{\mathrm{i}}_{\mathrm{WL}}\mathrm{j}}$ and ${\mathrm{X}}_{{\mathrm{i}}_{\mathrm{BL}}\mathrm{j}}$ represent the prospective installed capacities of technology j in the worst and best locations for technology j. It should be noted that the worst and best technologies, and the worst and best locations can be calculated using Equations (3), (4), and (5), (6) respectively.

Demand—supply constraints: The effective power demand is to be met by the existing plants (generation) and new RE generation. This necessitates the following set of constraints (stated in parts (a) and (b) provided below) for the prospective (to be installed) RE capacities.

a. Prospective off-grid installation constraints: The demand and supply constraints for off-grid installation locations are formed by using Equation (17).

$${\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{CF}}_{\mathrm{ij}}{\mathrm{X}}_{\mathrm{ij}}\ge {\overline{\mathrm{ED}}}_{\mathrm{i}}-{ \mathrm{PS}}_{\mathrm{i}}\hspace{1em}\forall \mathrm{i}$$

(17)

where, ${\mathrm{CF}}_{\mathrm{ij}}$ is the capacity factor in percentage (%) for technology j in location i, and i = 1…d, where d is the number of off-grid installation locations. ${\mathrm{PS}}_{\mathrm{i}}$ is the current power supply to each off-grid installation location i in $\mathrm{MW}$, and ${\overline{\mathrm{ED}}}_{\mathrm{i}}$ is the estimated demand in $\mathrm{MW}$ for each off-grid installation location i in order for the electricity access level to be 100%. Note that, ${\overline{\mathrm{ED}}}_{\mathrm{i}}$ can be calculated by using Equation (8).

b. Prospective on-grid installation constraints: To satisfy the on-grid demand and supply constraints, Equation (18) is formulated.

$${\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\displaystyle \sum }_{\mathrm{i}=\mathrm{d}+1}^{\mathrm{n}}{\mathrm{CF}}_{\mathrm{ij}}{\mathrm{X}}_{\mathrm{ij}}\ge \left(\mathrm{RPD}-\overline{{\mathrm{PD}}_{\mathrm{e}}}\right)-\left(\mathrm{RPS}-{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{d}}{\mathrm{PS}}_{\mathrm{i}}\right)$$

(18)

where, $\mathrm{RPD}$ and $\mathrm{RPS}$ are the reported power demand and total power supply in $\mathrm{MW}$ for the region of analysis respectively, and $\overline{{\mathrm{PD}}_{\mathrm{e}}}$ is the total estimated power demand of the off-grid installation locations in $\mathrm{MW}$. It should be noted that $\mathrm{RPD}$ and $\mathrm{RPS}$ are usually obtained from the system operator (SO), while $\overline{{\mathrm{PD}}_{\mathrm{e}}}$ can be calculated using Equation (7). Also, it is worth mentioning that RPS include the power supply of all locations (both on-grid and off-grid), while ${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{d}}{\mathrm{PS}}_{\mathrm{i}}$ is the total power supply for off-grid locations. Hence, the difference between these two parameters leads to the power supply for on-grid locations. Hence, the right-hand side of Equation (18) leads to the difference between demand and supply for on-grid installation locations.

• System reserve requirements: To guarantee reliability of the system, the system reserve (SR) must be satisfied. SR in percentage (%) can be defined as an additional and base component of load demand. Hence, the SR requirement is expressed by the following equations (Parts (a) and (b) shown below).

  a. 

  Prospective off-grid reserve constraints: The off-grid SR requirement can be satisfied by using Equation (19). It should be noted that the SR requirement is an additional amount of electricity that is used in supporting the actual load demand requirement. In most cases it is about 10% of the actual demand.

  $${\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{CF}}_{\mathrm{ij}}{\mathrm{X}}_{\mathrm{ij}}\le \left(1+\mathrm{SR}\right)\left[{\overline{\mathrm{ED}}}_{\mathrm{i}}-{ \mathrm{PS}}_{\mathrm{i}}\right]\forall \mathrm{i}$$

  (19)

  b. 

  Prospective on-grid reserve constraints: Equation (20) is used to satisfy the SR requirement related with on-grid installations.

  $${\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\displaystyle \sum }_{\mathrm{i}=\mathrm{d}+1}^{\mathrm{n}}{\mathrm{CF}}_{\mathrm{ij}}{\mathrm{X}}_{\mathrm{ij}}\le \left(1+\mathrm{SR}\right)\left[\left(\mathrm{RPD}-\overline{{\mathrm{PD}}_{\mathrm{e}}}\right)-\left(\mathrm{RPS}-{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{d}}{\mathrm{PS}}_{\mathrm{i}}\right)\right]$$

  (10)

• Area Constraints: The area required to install the RESs in a given location can be calculated by using Equation (20).

  $${\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{A}}_{\mathrm{ij}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{ij}}\le {\mathrm{AR}}_{\mathrm{i}}$$

  (21)

  where ${\mathrm{A}}_{\mathrm{ij}}$ is the area required (${\mathrm{km}}^2$) to install 1 MW of technology j in location i and, ${\mathrm{AR}}_{\mathrm{i}}$ can be calculated by using Equation (11).
• Natural constraints: the natural non-negativity constraints of the decision variables are given in Equation (22).

  $${ \mathrm{X}}_{\mathrm{ij}}\ge 0\hspace{1em}\hspace{1em}\forall \mathrm{i},\mathrm{j}$$

  (22)

3.3. Proposed Solution

The solution to the optimization problem developed in this paper, and hence the framework developed to incorporate all related parts, are described in this sub-section. It should be noted that the linear programming (LP) was used as a solution method to the optimization problem. LP entails minimization and maximization of a problem that has an objective function in a form of a linear function in the presence of linear inequality or/and equality constraints. LP is a mathematical technique for finding the optimum resources allocation and obtaining a given objective over a set of other alternatives [47]. Since 1947 when George B. Dantzig introduced simplex method, LP has been used broadly in engineering, industry, military, and government among others. The popularity of LP can be attributed to modeling large and complex problems, and the user ability to solve such problems in a considerable amount of time through the use of computer algorithms [48]. The optimization problem is solved by using MATLAB software. In MATLAB, linprog function is used as it is referred as one of the most recommended functions for solving such a problem. It is an extension package for the environment and language for statistical graphics and computing called R. Thus, linprog provides tools for linear optimization [49].

(1)

Construct the optimization problem by following the below-mentioned steps:

○

Use the prospective installed capacity for each distribution location (DL) as a decision variable.

○

Use the RESs potential data from NASA or any other database to determine the comparative potentials of off-grid and on-grid locations using Equations (1) to (6).

○

Get the access level data and power supply data for each location to estimate the total demand of off-grid locations by using Equations (7) to (10).

○

Use Equation (11) to calculate the landmass to be used for each RES.

○

Define the objective function using Equation (12).

○

Form the comparative constraints using Equations (13) to (16).

○

Form the demand and supply constraints for the off-grid and on-grid locations using Equations (17) and (18).

○

Form the SR constraints using Equations (19) and (20).

○

Form the area constraints using Equation (21).

○

Define upper and lower bounds for all variables using Equation (22).

(2)

Solve the linear optimization problem constructed in Step (1) with linprog function by using the parameter-equation sets described in

Table 2. Linear optimization approach: Parameters vs. related equations.

Input Parameter	$\mathbf{f}$	$\mathbf{A}$	$\mathbf{b}$	${\mathbf{A}}_{\mathbf{eq}}$	${\mathbf{b}}_{\mathbf{eq}}$	$\mathbf{u}\mathbf{b}$	$\mathbf{l}\mathbf{b}$
Related Equations	12	13–21	13–21	-	-	22	22

.

The function has the parameters explained below. The utilization of the function for this optimization problem is presented in Table 2.

Where $\mathrm{f}$ is the objective function, $\mathrm{A}$ is a z-by-v matrix, where v is the number of variables and z is the number of inequalities, $\mathrm{b}$ is a vector of length z, ${\mathrm{A}}_{\mathrm{eq}}$ is the matrix that summarizes all equality constraints, ${\mathrm{b}}_{\mathrm{eq}}$ is a vector of length u, $\mathrm{ub}$ represents the matrix of upper bounds applied to the variables and $\mathrm{lb}$ represents the matrix of lower bounds applied to the variables.

4. Case Study

4.1. Country Selection

Most of the developing countries have low electricity access levels and energy crisis due to generation, transmission and distribution problems [50]. In this paper, Nigeria is used as a case study in order to implement the proposed models mentioned in Section 3. The main reason for selecting Nigeria is the fact that the country is faced with severe electricity problems, which are deterring its development, despite the availability of enormous indigenous natural resources [16].

Table 3. Generation of electricity by power stations in Nigeria on 23/03/2017 [11].

Plant	Installed Capacity (MW)	Average Actual Generation (MWh/h)	Installed Capacity Utilization (%)	Generation Capacity (MW)	Generating Plant Contribution (%)	Peak Generation (MW)
Hydro Stations
Kainji G.S	760.00	376.58	49.55	440.00	6.47	398.00
Jebba G.S	578.40	324.46	56.10	380.00	5.59	367.00
Shiroro G.S	600.00	241.96	40.33	300.00	4.41	271.00
Sub-total	1938.40	943.00	48.65	1120.00	16.47	1036.00
Thermal Stations
G.S ST(1–5)	1100.00	308.25	28.02	660.00	9.71	348.00
Sapele Steam	720.00	0.00	0.00	0.00	0.00	0.00
Delta II–IV G.S	900.00	303.19	33.69	380.00	5.59	345.00
Afam IV and V	351.00	0.00	0.00	0.00	0.00	0.00
Geregu G.S	414.00	123.75	29.89	290.00	4.27	250.00
Omotosho G.S	335.00	207.33	61.89	304.00	4.47	221.60
Olorunsogo	335.00	172.92	51.62	266.00	3.91	186.30
Egbin G.S ST6	220.00	178.88	81.31	220.00	3.24	187.00
Sapele NIPP	500.00	159.08	31.82	225.00	3.31	212.10
Alaoji NIPP	500.00	113.17	22.63	360.00	5.30	117.20
Olorunsogo NIPP	750.00	4.80	0.64	360.00	5.30	0.00
Geregu NIPP	450.00	100.22	22.27	435.00	6.40	117.00
Omotosho NIPP	500.00	289.58	57.92	360.00	5.30	287.80
Ihovbor NIPP	500.00	108.75	21.75	225.00	3.31	114.10
Odukpani	500.00	140.24	28.05	120.00	1.77	80.50
Gbarain	120.00	104.30	86.92	115.30	1.70	115.30
Okapi G.S	480.00	314.67	65.56	450.00	6.62	328.00
Afam VI G.S	650.00	108.99	16.77	650.00	9.56	108.00
ASCO G.S	110.00	0.00	0.00	0.00	0.00	0.00
Ibom G.S	155.00	77.84	50.22	115.30	1.70	82.20
AES	294.00	0.00	0.00	0.00	0.00	0.00
Trans-Amadi	100.00	0.00	0.00	0.00	0.00	0.00
Rivers IPP	180.00	0.00	0.00	0.00	0.00	0.00
Omoku G.S	150.00	41.75	27.83	80.00	1.18	43.00
Paras energy	72.00	62.45	86.74	63.00	0.93	63.20
Total	12,324.40	3863.16	31.35	6798.60	100.00	4242.30

presents the power generation in Nigeria corresponding to 23rd March, 2017 [11]. The table shows that most of the generating plants in Nigeria are fossil fuel-based plants, and, most of them operate below their installed capacity. However, generation is not the only problem in the country’s electricity sector, as transmission and distribution also have problems. Identifying the significance of electricity access in economic growth and development in the country, a comprehensive reform was carried out in the sector [51,52]. One of the aims of the reform is improving electricity access and reliability across the nation with a view of achieving energy security in the shortest possible time. Nigeria, like any other country in sub-Saharan Africa, requires grid-extension to provide electricity access to areas with no electricity. However, grid extension needs large capital for high voltage, medium voltage and low voltage lines [53].

As mentioned by Iwayemi [54], the projected amount of investment to meet the power demand expansion of Nigeria in 2030 is estimated to be about $262 billion. However, such a high amount requires policy consistency, institutional framework, investment security, incentive structure, and, efficient planning of electricity generation, transmission and distribution. Moreover, numerous studies conducted in the past have revealed that decentralized electrification, whether stand-alone or mini-grid systems, is increasingly becoming sustainable and economically feasible option for electrifying areas where extending the grid may be physically impossible and expensive [50,55]. Hence, for Nigeria to improve its electricity access, energy planners must adopt the optimal design that involves off-grid and grid extension of electrification in a manner that increases RE utilization. However, policies and cost associated with RE installations posed barriers in the diffusion, development, and commercialization of RE in Nigeria. To this end, it is necessary to design cost-effective RE generation systems that can effectively reduce the installation costs, leading to a significant competitive edge for all stakeholders in the sector.

According to Nigeria Renewable Energy Master Plan (REMP), the country seeks to increase the supply of electricity from RESs to 36% by 2030 [56]. Also, the economic instrument policy that is in force in the country aims at encouraging the DLs to procure at least 50% of their electricity from renewables. Another target set in the country’s REMP is to increase the electrification rate from about 40% to 75% by 2025 [57]. However, for such a developing country to be self-sustainable, it is evident that electrification rate should be increased further. Therefore, in this paper, the proposed framework was applied to Nigeria DLs with the aim of deciding the amount of required RE installations to support the existing power capacity and hence to provide adequate power production while optimizing the use of RESs.

In this case study, the aim is to determine the total optimal cost of installing RE technologies in different DLs in Nigeria. The optimization is divided into three different scenarios. The first scenario (off-grid installation setup) optimizes (minimizes) the total cost of installing RESs for the DLs with electricity access level below 50%. Conversely, the second scenario (on-grid installation setup) evaluates the optimal cost of installing RESs for the DLs with electricity access level of 50% and above. The third scenario finds the optimal cost of installing off-grid RESs for all the DLs.

4.2. Data Collection

The modeling process employed several data, and this section will present an overview of the dataset used. In this case study, the number of adopted technologies are two; thus m is 2 (wind and solar technologies), while the number of locations, n, is 10. As shown in Figure 2, there are 11 DISCOs in Nigeria. However, it should be noted that since Eko and Ikeja DISCOs are within the same geographical location, they are treated as a single DL. The wind speed and solar radiation for each DL (as presented in Figure 3) are calculated by taking the averages of the wind speed and solar radiation data of different locations within that DL (obtained from NASA database and previous studies [17,39,40,58,59]). In addition to this, the electricity access levels for different states in Nigeria are presented in Figure 4. These were used to calculate the average access level in each DL and hence the preferred connection (or installation) type (i.e., on-grid or off-grid). The power supply to each DISCO was obtained from the National Control Centre Osogbo [60]. In addition, the SR constraints (Equations (19) and (20)) for both off-grid and on-grid plants were assumed to be 10%.

Compared to the rest of the world, there is a lack of RE related data in Africa [61], and, Nigeria is not an exception to this. The lack of RE cost data, specific to Nigeria, led to the adoption of the general costs from National RE laboratory listed in

Table 4. Installation costs and system sizes for photovoltaic (PV) and wind technologies [62].

Technology Type	Mean Installed Cost ($/kW)	Size (acres/MW)
PV < 10 kW	3897	3.2
PV 10–100 kW	3463	5.5
PV 100–1000 kW	2493	5.5
PV 1–10 MW	2025	6.1
Wind < 10 kW	7645	30
Wind 10–100 kW	6118	30
Wind 100–1000 kW	3751	30
Wind 1–10 MW	2346	44.7

. Table 4 not only presents the standard mean installed cost values ($/kW), but also shows the system size (acres/MW) required for wind and solar installations [62]. It should be noted that the cost of PV technology (reported in Table 4) includes the cost of module, inverter, structural balance of system (BOS), electrical BOS, and the install labor. Similarly, wind technology cost includes the cost of wind turbine, electrical infrastructure, installation, foundations and planning and development. For the off-grid systems, in case of both technologies, the system costs include the storage system. It should be noted that the data from this reference was last updated in 2016. The off-grid and on-grid technology sizes were selected to be 100–1000 kW and 1–10 MW respectively. These were used to determine the related costs. As shown in Table 4, the installation costs for off-grid solar and wind used in the study is $2493 and $3751 per kW respectively. Similarly, the on-grid solar and wind installation costs used in the study is $2025 and $2346 per kW respectively. Finally, the details of the modeling parameters used for each DL, demand and supply calculations using (Equations (7) to (10)), area calculations using (Equation (11)), and capacity factor calculations are presented in the next section.

Table 5. Descriptive statistics of wind speed and solar radiation data for each DISCO location (DL). CV—coefficient of variation; SD—standard deviation.

s/n	DL	Wind Speed (m/s) at 10 m Height					Yearly Solar Radiation (kWh/m2/day)
		Mean	Min	Max	SD	CV	Mean	Min	Max	SD	CV
1	Abuja	3.63	2.31	5.35	0.20	0.06	5.41	5.09	5.46	0.17	0.03
2	Benin	2.58	1.77	3.38	0.21	0.08	4.75	4.52	4.93	0.17	0.04
3	Eko and Ikeja	4.35	2.80	4.69	0.18	0.04	4.73	4.32	4.89	0.22	0.05
4	Enugu	4.27	2.08	5.75	0.15	0.04	4.50	4.23	4.91	0.12	0.03
5	Ibadan	4.12	2.26	5.04	0.19	0.05	4.40	4.15	5.89	0.26	0.06
6	Jos	4.83	3.09	9.47	0.15	0.03	4.51	4.22	5.76	0.14	0.03
7	Kaduna	5.13	2.31	7.21	0.19	0.04	5.87	5.61	6.23	0.29	0.05
8	Kano	5.45	3.19	9.39	0.21	0.04	5.82	5.58	6.15	0.11	0.02
9	Port Harcourt	3.86	2.08	4.60	0.13	0.03	4.92	4.12	5.04	0.14	0.03
10	Yola	4.88	3.18	5.25	0.18	0.04	5.92	5.69	5.98	0.17	0.03

shows the descriptive statistics that include the mean, minimum (min), maximum (max), standard deviation (SD) and Coefficient of Variation (CV) of wind speed and solar radiation of each DL. It should be noted that the table was constructed using the data available from NASA database and literature [17,39,40,58,59].

4.3. Calculation and Determination of the Modeling Parameters for Nigeria

In this sub-section, the novel equations developed in Section 3.1 are applied to Nigeria, and hence, the electricity access levels, preferred application types, comparative constraints, and capacity factors are determined. Also, the landmass required for RESs installations, best location for each technology and best technology to be installed in each DL are determined.

4.3.1. Electricity Access and Preferred Connection

Table 6. Average electricity access levels and preferred applications for DLs.

s/n	DL	Access Level (%)	Preferred Application
1	Abuja	56	On-grid
2	Benin	67	On-grid
3	Eko and Ikeja	99	On-grid
4	Enugu	65	On-grid
5	Ibadan	75	On-grid
6	Jos	34	Off-grid
7	Kaduna	41	Off-grid
8	Kano	36	Off-grid
9	Port Harcourt	61	On-grid
10	Yola	25	Off-grid

presents the average electricity access level and the preferred application type for each DL. The application type (on-grid or off-grid) was chosen based on the average electricity access level (as presented in Figure 4) of each location. The applications in locations with electricity access level at 50% or above were preferred to be on-grid while those below 50% were preferred to be off-grid (in the case of first and second scenarios). Actually, at this stage, it should be noted that 50% is just a threshold selected by the authors of this study, and hence it can easily be changed according to the requirements of the energy planners. This would change the type of the prospective installations. However, the methodology developed in this study is designed in a way to compensate such changes. It should be noted that the average electricity access levels reported in Table 6 are calculated by using the data illustrated in Figure 4. Moreover, the preferred application types seen in Table 6 are determined as explained in Section 3.

4.3.2. Required Landmass Calculation

The area required for RE installations for each DL can be calculated using Equation (11). The total area and area required for RE (wind and solar) installations in each DL are shown in Figure 5. The x-axis of the figure lists the DLs while y-axis shows the respective landmass. From Table 5, the values on the left y-axis represent ${\mathrm{A}}_{\mathrm{i}}$ (the total area available for installation in location i (each DL) as shown in Equation (11). It is clear from the figure that Yola has the largest area for RE installations, while Eko and Ikeja have the least available area for RE installations.

4.3.3. Capacity Factor Calculation

In this sub-section, the wind and solar related capacity factors (CFs) for each DL are presented.

Table 7. Wind and solar capacity factors (in percentages) for each DL.

s/n	DL	Wind Energy Capacity Factor (%)	Solar Energy Capacity Factor (%)
1	Abuja	17.50	15.09
2	Benin	10.00	14.60
3	Eko and Ikeja	23.00	15.30
4	Enugu	18.80	15.00
5	Ibadan	21.00	19.00
6	Jos	23.00	20.70
7	Kaduna	26.00	21.10
8	Kano	28.00	21.60
9	Port Harcourt	16.00	17.00
10	Yola	27.00	21.26

is constructed by using the data available in RETScreen and the data available in the literature [17,65]. The CF calculations for each DL were done by calculating the CF of all the available locations (within each DL) in RETScreen and finding the average. Also, due to lack of locations available in RETScreen, the CFs available in the literature [17] was averaged together with that of RETScreen. As shown in Figure 6, it is clear that wind has a better capacity factor when compared with solar in Nigeria. Also, Kano and Benin has the highest and lowest wind and solar capacity factors respectively.

4.3.4. Determination of the Potential Constraints

Table 8. Modeling parameters used for each DL.

s/n	DL	Wind Speed (m/s) at 10 m Height	Yearly Mean Solar Radiation (kWh/m2/day)	Comparative Wind Potential	Comparative Solar Potential
1	Abuja	3.63	5.41	0.81	1.06
2	Benin	2.58	4.75	0.57	0.93
3	Eko and Ikeja	4.35	4.73	0.97	0.93
4	Enugu	4.27	4.50	0.95	0.89
5	Ibadan	4.12	4.40	0.91	0.87
6	Jos	4.83	4.51	1.08	0.89
7	Kaduna	5.13	5.87	1.14	1.16
8	Kano	5.45	5.82	1.65	1.15
9	Port Harcourt	3.86	4.92	0.86	0.97
10	Yola	4.88	5.92	1.08	1.17

shows average RE potential, comparative wind potential and comparative solar potential for each location. The comparative potential for each RESs (wind and solar) was calculated by comparing the potential in each DL with the average, by using Equations (1)–(6) provided in Section 3.1. For example; from Table 8 the comparative wind and solar potentials (0.81 and 1.06) respectively, represents ${\overline{\mathrm{P}}}_{11}$ and ${\overline{\mathrm{P}}}_{12}$ respectively. Putting the comparative RE potential in Equation (2), the worst and best technology, worst and best location can be determined from Equations (3) to (6). It is important to note that n is equal to two and m is equal to 10 as Eko and Ikeja DISCOs are within same geopolitical location.

4.3.5. Demand and Supply Calculations

In

Table 9. Demand and supply calculations for each DL.

s/n	DL	Power Supply (MW)	Calculated Demand (MW)	Scaling Factor	Estimated Power Demand (MW)
1	Abuja	350.44	625.79	3.35	2096.39
2	Benin	272.84	407.22	3.35	1364.19
3	Eko and Ikeja	788.20	796.16	3.35	2667.14
4	Enugu	272.84	419.75	3.35	1406.16
5	Ibadan	394.09	525.45	3.35	1760.26
6	Jos	166.73	490.38	3.35	1642.77
7	Kaduna	242.52	591.51	3.35	1981.56
8	Kano	242.52	673.67	3.35	2256.79
9	Port Harcourt	197.05	323.03	3.35	1082.15
10	Yola	106.40	424.40	3.35	1421.74
Total	-	3033.33	5277.36	-	17,679.15

, the power demand and supply calculations for each DL are presented. The total estimated demand, the estimated power demand for each DL (when the access level is at 100%), and scaling factor were calculated using Equations (7)–(9) respectively. Although the formulation in this study is general, it should be noted that due to lack of information for the case study of Nigeria, the power supply to each DL data used in this study corresponds to 17 December, 2014, and it was obtained from the National Control Centre Osogbo [60]. The peak demand data (17,720 MW) corresponds to March, 2017 and it was obtained from Transmission Company of Nigeria (TCN) [11]. It is important to note that the scaling adopted in this study yielded a peak demand of 17,679 MW which is close to the estimated peak demand forecast of 17,720 MW.

5. Results

The proposed framework has been tested for Nigeria. MATLAB has been used to solve the linear cost minimization problem developed in the framework. Thus, the optimal RE installation capacities for each technology, and each location, have been determined in Section 4. Section 5.1 presents the simulation results of first scenario, which analyzes the off-grid setup. Section 5.2 provides the results of second scenario that evaluates the on-grid setup and Section 5.3 gives the simulation results of third scenario, which presents the results of off-grid setup for all the DLs.

5.1. First Scenario: Prospective Off Grid Setup

In this scenario, all the DLs with electricity access level below 50% are taken into account in order to determine the optimal RESs allocation to minimize the total cost. The optimal allocation refers to the optimal capacities to be installed for each technology in each location (prospective installed capacities). The simulation results, mainly the prospective installed capacities, are presented in

Table 10. Optimal capacities and generation for off-grid setup.

DL	Prospective Total Installed Capacity (MW)	Prospective Installed Solar Capacity (MW)	Prospective Installed Wind Capacity (MW)	Average Power Generation (MW)
Jos	7444.65	3853.28	3591.37	1623.64
Kaduna	7721.45	4130.08	3591.37	1912.94
Kano	8542.08	4130.08	4412.00	2215.69
Yola	7721.45	4130.08	3591.37	1851.65

. The optimization results for off-grid setup indicate that in Jos, solar should be favored, as the optimization results show that 51.76% of the installations should be solar while wind’s share is 48.24%. Also, the solar and wind shares in Kaduna are found to be 53.49% and 46.51% respectively. Similarly, for Kano, the optimization revealed that 48.35% and 51.65% of the installation is assigned to solar and wind respectively. For Yola, solar and wind shares represent 53.49% and 46.51% of the installations respectively. It should be noted that in this scenario, the main aim was to supply the demand-supply gap of each DL, by using the prospective installations only in that DL. The prospective total installed capacity of all the off-grid setup DLs was found to be 31,429.63 MW while the total average power generation was found to be 7603.92 MW. Furthermore, the optimal total cost of installing RESs in all the off-grid setup DLs is found to be $97.46 billion.

5.2. Second Scenario: Prospective On Grid Setup

The problem setup in this scenario makes sure that the optimal RE capacities are determined for each on-grid setup DL. It should be noted that for on-grid setup, since the DLs have substantial level of electricity access, the installation does not have to cover the supply and demand gap at each location. Only the total gap should be supplied. As presented in

Table 11. Optimal capacities and generation for on-grid setup.

DL	Prospective Total Installed Capacity (MW)	Prospective Installed Solar Capacity (MW)	Prospective Installed Wind Capacity (MW)	Average Generation (MW)
Abuja	2256.89	2256.89	0	3404.82
Benin	1427.28	1427.28	0	208.38
Eko and Ikeja	3254.20	1627.10	1627.10	662.23
Enugu	2724.81	1427.28	1297.53	445.05
Ibadan	3054.38	1427.28	1627.10	580.05
Port Harcourt	22,564.89	22,564.89	0	3610.38

, the optimization results show that Abuja and Port Harcourt are allocated 81.18% (40.59% each) of the proposed installed capacity.

Benin, Enugu, and Ibadan are allocated 2.57%, 4.90%, and 5.50%, while Eko and Ikeja is allocated 5.85% of the prospective installed capacity. The prospective installed wind capacity for Abuja, Benin and Port Harcourt are zero due do the poor wind potential in those DLs. Thus, the prospective total installed capacity of all the on-grid setup DLs was found to be 55,590.45 MW while the total average power generation was found to be 8910.91 MW. It is important to note that the results are in line with the target stated in the country’s REMP, which states that 50% of each DISCO’s purchase should come from renewables. Furthermore, the total cost of installation is found to be $114.03 billion.

5.3. Third Scenario: Prospective Off Grid Setup for all the DLs

In this scenario, all the DLs are assumed to be off-grid in order to determine the optimal RESs allocation, and minimize the total cost. The simulation results, mainly the prospective installed capacities, are presented in

Table 12. Optimal capacities and generation for off-grid setup.

DL	Prospective Total Installed Capacity (MW)	Prospective Installed Solar Capacity (MW)	Prospective Installed Wind Capacity (MW)	Average Generation (MW)
Abuja	11,789.75	6155.43	5634.32	1920.55
Benin	9760.08	5192.36	4567.72	1200.49
Eko and Ikeja	8495.19	4757.31	3737.88	1626.83
Enugu	7276.63	3485.51	3791.12	1246.65
Ibadan	7513.95	3839.62	3659.33	1502.79
Jos	7444.65	3853.28	3591.37	1623.64
Kaduna	7721.45	4130.08	3591.37	1912.94
Kano	8542.08	4130.08	4412.00	2215.69
Port Harcourt	5900.67	3854.65	2046.02	973.61
Yola	7721.45	4130.08	3591.37	1851.65

. It should be noted that in this scenario, the main aim was to supply the demand-supply gap of each DL, by using the prospective installations only in that DL. The optimization results for all-off-grid setup indicate that in Abuja, solar should be favored, as the optimization results show that 52.2% of the installations should be solar while wind share is 47.8%. Also, solar and wind shares in Benin are found to be 53.2% and 46.8% respectively. For Eko and Ikeja, the optimization revealed that 56% and 44% of the installation is assigned to solar and wind respectively. For Enugu, solar and wind shares represent 47.9% and 52.1% respectively. For Ibadan, the results show that 51.09% and 48.91% of the installation is assigned to solar and wind respectively. Finally, in Port Harcourt, 65.33% and 34.67% of installations should be allocated to solar and wind respectively Also, Jos, solar should be favored, as the optimization results show that 51.76% of the installations should be solar while wind’s share is 48.24%. Also, the solar and wind shares in Kaduna shares represent 53.49% and 46.51% respectively. Similarly, for Kano, the optimization showed that 48.35% and 51.65% of the installation is allocated to solar and wind respectively. For Yola, solar and wind are found to be 53.49% and 46.51% of the installations respectively. It should be noted that in this scenario, the RE installation is designed in such a manner that each DL can generate its own power. Furthermore, the prospective total installed capacity of all the off-grid setup DLs was found to be 82,165.90 MW while the total average power generation was found to be 16,074.84 MW. Furthermore, the optimal total cost of installing RESs in all the off-grid setup DLs is found to be $244.33 billion.

6. Sensitivity Analysis

Sensitivity analysis is a method used to evaluate how a change in a given parameter can affect the performance or outputs of the system. It can be applied to explore the accuracy and robustness of the model results under uncertain conditions [66,67]. It should be noted that a parameter (sensitivity variable) is any input data that is not a decision variable. In order to determine the model robustness, sensitivity analysis is needed. The sensitivity analysis performed in this paper, was carried out by varying the installation costs of both on-grid and off-grid scenarios, in the following manner; the sensitivity on the total installation cost is based on wind and solar technologies’ installation costs.

Table 13. Sensitivity variables and corresponding values.

Sensitivity Variable	Installation Cost ($/kW)		SD of Installation Cost (%)
	On-Grid	Off-Grid
Solar	2025	2493	±10, ±20, ±30
Wind	2346	3751	±10, ±20, ±30

shows all the associated cost values and sensitivity analysis related cases. The cases are: +10%, +20%, +30%, −10%, −20%, and −30% change in wind technology installation cost only (keeping solar related cost constant), and solar technology installation cost only (keeping wind related cost constant), creating 12 different cases for both on-grid and off-grid scenarios. Hence, totally there are 24 scenarios. It should be noted that the percentage changes mentioned above represent the SD of the installation costs.

Figure 6 and Figure 7 present the sensitivity analysis of change in technology related costs and total installation costs for off-grid and on-grid scenario respectively. The results indicated that decrease in the wind technology costs will significantly reduce the total installation cost in the case of off-grid system (rather than the case of having a decrease in solar energy related costs), while the on-grid total installation costs can be decreased more when the cost associated with installing solar technology is reduced (rather than the case of having a decrease in win energy related costs).

7. Discussion

The results presented in previous section show that wind is not viable in some parts of the country; this is due to the low potential of wind energy especially in the southern part of the country. This can be justified from the study conducted by [41,68]. The results of the sensitivity analysis reveal that the cost associated with on-grid RE systems can be reduced considerably by reducing the installation cost of solar energy technologies. This in fact requires considering more and effective policies that will aim at promoting solar technology. Conversely, for off-grid system, the analysis shows that the government and other stakeholders in the energy sector should focus more on policies that will aim at reducing the cost of installing wind energy technology. According to [69] private sector investments for RE have been inadequate. This is mainly due to lack of supportive policies and enabling environment in Nigeria. The government of Nigeria must identify and develop clear policy incentives for increased private sector participation in the delivery of RE. This is because the government alone cannot sufficiently fund such projects that are capital intensive. The reasons behind this claim can be explained as follows: As mentioned in the International Monetary Fund (IMF) report 2014 [70], the country’s oil and gas industry account for about 95% and 75% of total export and government revenue respectively, making the economy vulnerable to volatility of gas and oil prices. Example, a quick transition from an era of flourishing income when a barrel of crude oil was sold at $112.7 in December 2013, to the era when a barrel of crude oil was sold at $30 in 2016 [71]. This change in the price of crude oil has negatively affected the country’s economic growth where in some cases the country finds it even difficult to implement its budget. In light of the above-mentioned circumstances, the government has to enact more and effective RE policies that will support private investors and hence improve the development and deployment of RE. However, these suggestions are not specific to Nigeria, but cover many other developing countries.

For developing countries to improve their electricity access, security and sustainability, RE utilization has to be encouraged through proper planning (Supply-Side Management (SSM)), and implementation of effective policies. Optimization (which is used to develop the framework in this paper) is seen as one of the methods used to achieve proper planning in electricity sector, especially when dealing with RE installations. Unlike other electricity SSM methods (such as economic dispatch, unit commitment, and system sizing) focusing on cost minimization of power generation of the individual installations (requiring hourly load profile data), the novel framework presented in this study focuses on finding the minimum cost of installing RE technologies in large scale and total RE capacity to be installed in a region or country. Hence, this study is not limited to any specific installation type or size. It does not require the knowledge of system characteristics of the individual installations. Rather, it focuses on the total capacity that should be added to the electricity mix, without restricting the type and the size of the individual applications. For example, as seen in Table 11, in Enugu DL, 1427.28 MW and 1297.53 MW should be installed for solar and wind technologies respectively. In doing these installations, the government (or the stakeholders) may decide on having several power plants with higher capacities, or having even more power plants with less capacities. Also, the type of technology can be selected based on the government’s or the stakeholders’ decisions. Hence, it should be distinguished from other SSM related studies, as it can serve as a guide to the stakeholders (government) in setting RE targets and enacting RE related policies. This is mainly due to the fact that it can be used to determine the technologies that are the most viable in different locations in a country or region, along with the total capacities that need to be installed to improve the supply in those locations and reduce the gap between demand and supply in the whole country or region.

8. Conclusions

The framework and models described in this paper were designed to be flexible enough to be adjusted for use in different countries or regions. It thus allows for combining existing generation capabilities with increased use of RES; such as wind and solar, as used in the case study. The proposed optimization model was developed based on linear programming. The minimum costs of installing RES for three different scenarios (off-grid, on-grid and all-off-grid) were determined. It can be observed that the proposed models were successful in minimizing the total installation costs of RE (wind and solar) by favoring the best RE technologies in each location, and avoiding extra installations. The off-grid installation scenario results show that the optimal total installed capacities for related DLs, such as Jos, Kaduna, Kano, and Yola are 7444.65 MW, 7722.45 MW, 8542.08 MW, and 7721.45 MW respectively. Conversely, in the on-grid installation scenario, the optimal installations for each DL are found to be 22,564.89 MW, 1427.28 MW, 3254.20 MW, 2724.81 MW, 3054.38 MW, and 22,564.89 MW for Abuja, Benin, Eko and Ikeja, Enugu, Ibadan, and Port Harcourt respectively. Furthermore, the total installation costs of the first and second scenarios (that include having off-grid installations for some DLs and having on-grid installations for other DLs) are $97.46 billion and $114.03 billion respectively, with a total cost of $211.49 billion. On the other hand, the result of third installation scenario (only off-grid installation for all DLs), is found to be $244.33 billion. As evident from the results, incorporating RE in Nigeria’s power generation is vital in resolving the current power problem. Furthermore, building solely more on-grid renewable power plants is not enough; the system needs to be supported with off-grid renewable plants to curb the transmission constraints. Furthermore, the study shows that if the proposed framework is applied in Nigeria, it can help the country to achieve the RE targets stated in the country’s REMP.

For future work, the study can be improved by adding more constraints due to the existence of RE policies (such as capital incentives, tax credits, and green certificates). Also, the costs of adding transmission and distribution lines can be added to the framework. Furthermore, techno-economic analysis (that include simple payback period or internal rate of return) can be added to improve the study.