Exploring the boundaries of Solar Home Systems (SHS) for off-grid electrification_ Optimal SHS sizing for the multi-tier framework for household electricity access

With almost 1.1 billion people lacking access to electricity, solar-based off-grid products like Solar Home Systems (SHS) have become a promising solution to provide basic electricity needs in un(der)-electrified regions. Therefore, optimal system sizing is a vital task as both oversizing and undersizing a system can be detrimental to system cost and power availability, respectively. This paper presents an optimal SHS sizing methodology that minimizes the loss of load probability (LLP), excess energy dump, and battery size while maximizing the battery lifetime. A genetic algorithm-based multi-objective optimization approach is utilized to evaluate the optimal SHS sizes. The potential for SHS to cater to every tier of the Multi-tier framework (MTF) for measuring household electricity access is examined. The optimal system sizes for standalone SHS are found for different LLP thresholds. Results show that beyond tier 2, the present day SHS sizing needs to be expanded significantly to meet the load demand. Additionally, it is deemed untenable to meet the electricity needs of the higher tiers of MTF purely through standalone SHS without compromising one or more of the system metrics. A way forward is proposed to take the SHS concept all the way up the energy ladder such that load demand can also be satisfied at tier 4 and 5 levels.


Introduction
Almost 1.1 billion people lacked access to electricity in 2016 [1]. Most of these regions fortunately fall in latitudes that receive abundant sunshine. Additionally, as grid extension is not an immediate solution in most of the unelectrified regions, solar-based products like Solar Home Systems (SHS) have become an important stopgap solution.
An SHS is usually defined as a solar PhotoVoltaic (PV) generator rated 11-20 Wp (for entry level SHS) to more than 100 Wp (high power SHS) and a suitable battery storage [2]. The term Solar Home Systems may be used interchangeably with a standalone PV system, although the term has largely come to be used in the context of off-grid electrification.

Multi-tier framework (MTF) for household electricity access
Traditional view of looking at the electricity access as have/havenot condition is severely restricting. In reality, having access to a certain level of electricity (for example lights and mobile phone charging) leads to improved living conditions, thereby necessitating higher electricity needs in the future [3]. To capture the multi-dimensional nature of electricity access, a multi-tier framework (MTF) was proposed by [4]. Table 1 captures the MTF as described in [4]. Climbing up the so-called electrification ladder can therefore also be viewed in terms of moving up the tiers within the MTF. Although the notion of MTF is independent of the type of energy source that enables electrification, in this study, we consider solar PV being the only energy source and battery being the energy storage. In this paper, an optimal sizing methodology is introduced to optimally size SHS for every tier of the MTF, which gives insights on the level of PV and battery storage needed to enable electrification across the various tiers.

Importance of optimal SHS sizing
SHS sizing comprises the PV sizing, battery sizing, and the sizing of the power converters. PV sizing mainly depends on the total energy needed from the PV generator, which in turn depends on the load profile. A lower than adequate PV size results in system failure or a high number of loss of load events, i.e., high loss of load probability (LLP). LLP is a system metric that quantifies the system's reliability in meeting the load demand, as explained in detail in Section 2.1. A higher than adequate PV size, however, increases the amount of energy dumpedwhen the load is satisfied and the battery is full.
Power converter sizing mainly depends on the peak power of the PV and the load. Some degree of dimensioning flexibility can be achieved depending on the undersizing of the converter. A lower than peak power of the converter may not be suited for peak power operation but can perform more efficiently at most of the other operating points. This depends mainly on the frequency of occurrence of the different power levels expected throughout the operation of the converter. Thus, a suitably lower than peak sized converter can lead to cost savings. Compared to PV and power converter, however, battery sizing is far more involved.
The battery is a vital component of the SHS that not only enables energy storage of the PV output but also caters to the load when there is no solar generation. However, the battery is the most expensive SHS component while suffering from low lifetimes as compared to other SHS components. Additionally, a smaller than adequate battery size will result in failure to meet the load requirements (high LLP), while an oversized battery will drastically increase the upfront costs of the system. Also, a larger battery size can lead to lower Depth of discharge (DOD) levels, and therefore higher lifetime, whereas a smaller battery size can lead to lower lifetimes due to the deep DOD levels [5,6]. Usual battery lifetimes are much lower than typical PV module lifetime of 25 years. Therefore, a higher battery lifetime is advantageous as it means fewer replacements during the SHS lifetime. Battery costs and lifetime thus have an intricate relationship, making battery sizing extremely relevant albeit challenging in SHS design. Battery sizing and lifetime can, therefore, be considered as critical parameters when dimensioning an SHS.
An optimal SHS size can thus be considered as one that results from an SHS dimensioning exercise that minimizes the LLP, energy dump, and battery size while maximizing the battery lifetime.

Literature study
In general terms, most sizing-based studies utilize sizing criteria that result in a good trade-off between the system reliability (or power availability) and system investment cost. Previous studies have dealt with the sizing problem by only looking at one or two metrics, including LLP [7,8], system cost, and battery lifetime [9]. LLP has been considered as an objective in [10] as well as a constraint. In [11], the main objective function was to minimize the total SHS investment cost, while keeping LLP ⩽ 2%. LLP was also used as the primary sizing criterion for an off-grid PV-battery system in Bolivia [12], where the system size was determined for three different case studies: a household, school, and health center.
Additionally, battery lifetime has been included as a key parameter in a single weighted objective optimization for a PV array, diesel generator and battery system to minimize battery degradation and fuel consumption [13]. Similarly, the optimal sizing of a PV-battery-diesel generator hybrid system has been investigated [14], paying special attention to system cost and environmental impact of the off-grid system. In this case, the levelized cost of electricity and the carbon footprint of energy are defined as the main metrics that must be minimized.
Other sizing methods are based on intuitive "rules of thumb". The concept of days of autonomy (DOA) for battery sizing, which consists of finding the storage size that can fulfill the load for a predefined amount of days in the absence of solar generation is an example [15]. Another example is the concept of nights of autonomy (NOA). Both the DOA and NOA concepts are discussed in Section 2.2. More complex optimization methods have been introduced over the past years, especially iterative optimization approaches where the system performance for the objective is iteratively evaluated across the decision variable space [16,17]. For instance, a standalone PV-battery system has been dimensioned based on loss of power probability and life cycle cost using an iterative procedure [18].
However, the main drawback of these techniques is that the system is only optimized based on one objective function. Therefore, a system with multiple trade-offs, or objective functions, has not been completely tackled, especially for the application of solar home systems. In comparison, multi-objective optimization (MOO) techniques offer better applicability. Nowadays, MOO techniques based on artificial intelligence are widely used, which are generally more robust, and are better equipped to deal with multi-objective optimization problems [19]. Among them, Genetic Algorithms (GA) are powerful meta-heuristic techniques that are capable of reaching global optima with high accuracy and appropriate computational speed [20]. Former studies using MOO techniques in PV systems show different focus areas compared to our study. In [21], authors quantify the trade-offs between economic and environmental performances of rural solar PV projects. Authors in [22] use a sizing algorithm based on Particle Swarm Optimization (PSO) for a PV-battery-hydrogen fuel cell-based hybrid system. There is a clear research gap in literature with respect to studies focusing on optimal sizing of present-day SHS, especially taking into account battery lifetime as one of the objectives. In this study, the different SHS parameters need to be optimized simultaneously, making the MOO a necessity. Consequently, in this study, we use a GA algorithm for the task of optimal sizing of an SHS for different electrification levels defined by the MTF. In this optimization, the PV rating, battery capacity, and converter sizes are obtained taking into account the most optimum combination of LLP, excess energy, battery size, and battery lifetime.

Contributions of this paper
Following are the main contributions of this paper.
1. An optimal sizing methodology for SHS is presented that minimizes the LLP, excess energy, and battery size while maximizing the battery lifetime specific to SHS applications. 2. For the first time, optimal SHS sizing is investigated for the various tiers of MTF for household electricity access. 3. Inadequacies of standalone SHS are highlighted for higher tiers of electrification and a possible alternative is proposed for climbing the so-called electrification ladder.

System metrics
The system metrics used in this work are discussed below.

Loss of Load Probability
The Loss of Load probability is a measure of the system downtime. It is defined as the ratio of the amount of time the system fails to deliver the demanded power to the total amount of operation time the system was designed to deliver power for [8]. For modelling and simulating off-grid systems, this is an application specific or a user-defined constraint. For e.g., in [8], the LLP over a year was constrained to 1.8%, and in [23], a similar metric had a minimum limit of 2%.
where t i downtime, takes a value of 1 if the system fails to deliver the expected load demand in the ith time interval, and 0 if the system fully meets the load requirement; N is the time period of interest. Since the system is modelled with a 1-min data resolution, t i downtime, is updated every minute, while a 1-year long period of interest yields an N value of 525600.
The choice of LLP can also be dependent on the application. For instance, [15] states that the recommended LLP values for domestic illumination, appliances, and telecommunications applications are 0.01, 0.1, and 0.0001 respectively.

Unsatisfied load energy (E fail )
E fail quantifies the unmet energy demand in kWh over a given period of time. In this study, it is mathematically defined as the summation of the unsatisfied energy over a year for each time interval i, as shown in Eq. (2).
2.1.3. Energy dump (E dump ) and dump ratio (R dump ) This is the total amount of energy that is unused when the local battery is full and the load demand is met while the PV can still generate more power. In this study, this value is considered over a period of 1 year for the system. In order to make relative comparisons easier, a term dump ratio is introduced, which is the ratio between the total energy dump of the system in a year divided by the annual load energy need, as seen in Eq. (3).

SHS size
SHS size corresponds to the electrical dimensions of PV, battery and the power converters. Typically, this is specific to the rated power of the PV in Wp, total battery capacity in Wh, and peak power rating in W of the converter.

Battery lifetime
This is the lifetime in years of operation after which the battery capacity reduces to below 80% of its nominal rated capacity. At the end of this period, the battery is said to have reached its end of life (EOL).

System parameters
The following are the other important system parameters that are referred to in this paper.

State of charge and depth of discharge
The state of charge (SOC) refers to the battery charge as a fraction of the nominal capacity, while the depth of discharge (DOD) refers to the capacity discharged as a fraction of the initial capacity [5].

State of health
State of Health (SOH) is indicative of the fraction of the nominal rated battery capacity actually available for cycling.

Days of autonomy (DOA)
Days of autonomy refers to the number of days a particular battery size is capable of powering the full load demand assuming there is no PV generation. This is often used in approximate battery sizing. For e.g., a battery size in Wh can potentially be written as shown in Eq. (4).
where E load is the load energy requirement over a day, n DOA is the number of days of autonomy, batt is the average battery efficiency, and DOD is the average DOD of the intended battery cycling. Practitioners often use DOA or NOA to size the battery as a "rule of thumb". However, this can lead to inaccurate and suboptimal system sizing, as shown later in Section 4.1.

Nights of autonomy (NOA)
Nights of autonomy refers to the number of nights (night = nonsunlight hours of the day) where a battery can completely power the load demand assuming no PV generation to recharge the battery. This is also used in approximate battery sizing. For e.g., a battery size in Wh can potentially be written as shown in Eq. (5).
where E nightload is the load energy requirement in the non-sunlight hours in a day, and n NOA is the number of nights of autonomy.

Methodology
The various steps employed in the methodology of this study are described in this section. Section 3.1 presents the inputs to the SHS model, Section 3.2 presents the architecture of the system, Section 3.3 describes a dynamic PV output modeling method, Section 3.4 describes the steps involved in battery lifetime modelling, Section 3.5 discusses the power management scheme used in the standalone SHS, Section 3.6 describes the steps used to evaluate the converter rating, and Section 3.7 details a genetic algorithm (GA)-based multi-objective optimization (MOO) approach for SHS sizing.

Location and meteorological inputs to the SHS model
Since the ground level meteorological data was not readily available for the remote rural areas, the data for an Indian city (with 5 Equivalent sun hours of irradiance per day) was taken from the Meteonorm software database [24]. The meteorological data is assumed to be representative of other remote areas in similar latitudes in the tropical regions. The data used in this study has a resolution of 1 min.

Load profiles for the MTF
Stochastic load profiles were constructed for the various tiers of the MTF in a previous work by the authors [25]. Fig. 1 shows a sample load profile from a representative day for a tier 2 household. The householdlevel load datasets for all the tiers have been obtained from [25]. Table 2 shows the important load profile characteristics for each tier of the MTF, as obtained in the previous work [25].

Modular SHS architecture
Climbing up the so-called rural electrification ladder requires expansion of the off-grid system to cater to increased load demand. A modular architecture for SHS is therefore proposed that could potentially allow for expansion of the system at the household level. Fig. 2 shows the modular architecture.
All the SHS components, viz., PV, battery, and DC loads, are connected to the central DC bus via converters. Expansion of the system can be achieved by adding more of the SHS components to the DC bus via a converter, as shown in Fig. 2 with the dashed battery at the bottom. The DC bus can be operated at 12 V, 24 V or 48 V. As more high power loads are connected, a higher bus voltage is recommended to keep the current and therefore cable losses low.

Dynamic PV output
The dynamic PV output for each minute is calculated based on an elaborate PV model as illustrated in Fig. 3.
There are 4 different kinds of inputs used. Firstly, the location coordinates in terms of latitude and longitude are used to determine the sun position throughout the year. The two main parameters that quantify the sun position are the sun Azimuth (E s ) and the sun Altitude (A s ). Secondly, the ground level irradiance data for the location is used, as mentioned in Section 3.1.1. Along with E s and A s , the PV module orientation optimization model uses G G , DNI DHI , and G GHI to evaluate the module azimuth and tilt that maximizes the plane of array irradiance (G POA ). Additionally, the G POA is computed for the optimal azimuth and tilt.
The dynamic efficiency ( dyn ) of the PV module can be quite The total energy consumption for this day is 257 Wh with a peak of around 49 W. Data sourced from [26].

Table 2
Main load profile parameters: maximum peak power P max , minimum peak power P min , and average daily energy E daily for each tier based on the 1-year generated load profile [25]. different than its rated efficiency ( STC ). As dyn depends on both the module temperature (T mod ) and G POA , the module temperature needs to be estimated. The fluid dynamic model is used to estimate T mod by using T amb , wind speed (v w ), and the calculated G POA as inputs [15]. Finally, the PV efficiency is corrected and the dynamic efficiency dyn is calculated using the module area (A PV ), the temperature coefficient of power for the PV module (k), and STC . The selected PV module is Jinko Solar JKM265P. Although the rated power is 265 Wp, a normalized dynamic PV output scaled down to 10 Wp is considered for the optimal sizing methodology. The detailed methodology for modelling the dynamic PV output has been described in a previous work done by the authors in [8].

Estimating battery lifetime
In this study, a valve regulated lead acid (VRLA) battery is considered as the energy storage device. A practical battery lifetime estimation method is considered specific to SHS applications, wherein the battery ageing estimation is based on a dynamic capacity fading model introduced by the authors in [5]. The model uses battery manufacturers' datasheets to refer to the cycle life curves at different temperatures [27]. Constant round-trip efficiency of 90% is considered for the VRLA battery [5].
In the dynamic capacity fading model, micro-cycles of battery activity are considered based on the zero-crossings of the battery current. At the start of the simulation, the battery is assumed to be at 100% SOH. For each micro-cycle, the average DOD (DOD i ), temperature (T i ) and total energy throughput (E thri ) are evaluated. The proportional number of cycles spent under the same DOD and temperature levels, , is then evaluated, as shown in Eq. (6) [5]. The partial damage incurred by the battery (D i ) in the ith micro-cycle is calculated as shown in Eq. (7) [5].
This damage is then scaled and subtracted from the present SOH, so that when the cumulative damage becomes 1, the SOH reaches 80%. The SOH is therefore dynamically updated as the battery capacity fades with consecutive cycles. When the SOH falls below 80%, the simulation ends. The time at which the simulation ends is the battery lifetime (L) in years.
It should be noted that the lifetime estimation methodology only accounts for the cyclic ageing based on the application usage. Additionally, as the cycle-life curves are taken from the manufacturer's datasheets, choice of a different battery product or type will impact the lifetime calculations.

Power management scheme for standalone SHS
The algorithm used for power management in every time step (1 min) is shown in Fig. 4. E batt refers to the battery capacity in Wh in any time step.
As shown in the flowchart, at the start of every time step t, the excess power or the load deficit is computed via: , LLP = 1 for the given time step. It must be noted that a C-rate limit of 5C was imposed on the battery operation. This process repeats for every time step until the one year simulation is over, corresponding to 1 year of the load data and meteorological data.

Converter rating
The power converters shown in Fig. 2 need to be appropriately rated. Power rating of each converter is chosen depending on its intended application as seen below.

PV converter sizing
The PV converter optimal size was selected based on the sizing ratio R S defined in Eq. (9).
where P PV,Peak is the peak PV power and P Nom is the rated converter power.
In the studies conducted in [28,29], the effect of varying R S on the total yearly energy output from the PV converter was analyzed. The PV array output depends on the geographical location, and rarely has an output that is equal to or larger than its rated peak power. Hence, having = R 1 S or lower for a relatively small time during the year is unnecessary. In this study, the same approach for finding the optimal R S was performed. The yearly energy yield from the PV module was obtained for values of R 0.5 1. 5 S as shown in Fig. 5. The figure shows that for = R 0.9 S , the maximum energy yield is obtained. However, for = R 1.27 S , 96% of the maximum yield is obtained. Hence, for an around 41% reduction in the sizing ratio (from 0.9 to 1.27), only 4% of the energy yield is lost. Therefore, an R S of 1.27 was selected as the optimal sizing ratio to obtain the PV converter rating P Nom,conv , as it is the highest sizing ratio (and therefore most sizing gains) that guarantees more than 95% of maximum achievable yield. Moreover, past studies on solar converter sizing have concluded that a converter could be undersized by up to 30% of the PV array W p , as undersizing causes an insignificant reduction ( 5%) in the total yearly energy yield compared to the reduction in its cost [28,29].

Load converter sizing
The load converter was sized according to the peak load for each tier. Hence, for the load converter, = P P Nom max , which are the values found in Table 2.

Battery converter sizing
The battery converter is a bidirectional converter that processes power by charging or discharging based on the excess PV generation or greater load demand respectively. Hence, it should be sized appropriately to allow the maximum net charge or discharge power to go in/ out of the battery. The P Nom for the battery is then the maximum between the P deficit and P excess values.

Multi-objective optimization for standalone SHS sizing
Optimal SHS sizing is a complex task that brings to the fore the intricate interplay between the different SHS component sizes and the various system metrics described in Section 2.1. For e.g. a larger battery size can lead in general to longer battery lifetimes (at the cost of increased initial investments), while a smaller battery size will result in loss of load events (high LLP) as there won't be enough stored energy to power the load in the non-sunlight hours. Similarly, a lower than adequate PV size will result in high LLP. However, an indiscriminate increase in PV size will result in high E dump values, which is a waste of energy and should be avoided as much as possible. While an increase in PV or battery size increases the initial system costs, a high E dump value increases the levelized cost of kWh of the system.

Decision variables
For the optimal SHS sizing, 3 variables determine the performance of the system based on the various metrics. These are: PV size (Wp), battery size (Wh), and converter ratings (W). However, as seen in Section 3.6, the converter rating is dependent on the PV size and the load. The load profile is a given for the specific SHS application (MTF tier-based usage). Therefore, the primary independent decision variables used in this study are: (1) the PV size (Wp) and (2) the battery size (Wh).

Objectives
The main objectives can be written as (a) to minimize total cost (b) to minimize LLP and (c) to minimize E dump (or R dump ). However, the total system cost can be considered as a sum total of PV cost, battery cost, and the converter cost. Each of these costs are directly proportional to the rating of the SHS components. Additionally, the battery costs consist of both the initial costs and the replacement costs, owing to the lowest lifetime amongst the other SHS components. As the replacement costs go down with increasing battery lifetime, minimizing the total battery costs is the same as minimizing the battery size while maximizing the battery lifetime (although battery size and lifetime are not mutually independent variables).
The converter sizes are again dependent on the PV and load and therefore cannot be independently minimized to lower the costs. PV size is in a way already reflected in minimizing the dumped energy (E dump ). Therefore, the objective of minimizing total SHS costs can be crystallized down to the objectives of minimizing battery size, maximizing battery lifetime and minimizing E dump . This adaptation of the cost objective into battery size and lifetime helps in avoiding the actual costs in $ or €, for example, and maintains the generality of the methodology and the results.
Consequently, the following objectives are used for multi-objective optimization in this study.
(1) to minimize the battery size (2) to maximize the battery lifetime (3) to minimize the LLP and (4) to minimize the R dump .

Constraints
Constraints are necessary in the MOO process to curtail optimization run times and eliminating unwanted navigation of the algorithm employed within the search space. In this study, constraints were placed on the LLP and the E dump . Although LLP and E dump form part of the objective set, the additional constraints augment the efficiency to the optimization computation. The constraints used in the MOO can be stated as:

Genetic algorithm-based MOO
For performing a multi-objective optimization (MOO), the genetic algorithm toolbox of MATLAB was utilized. The gamultiobj function within the GA toolbox of MATLAB is based on a Non-dominated Sorting GA (NSGA)-II variant [30]. Fig. 6 outlines the steps contained in the GA-based MOO used in this study. The functioning of the genetic algorithm can be explained as follows.
1. Initialization. The initialization step involves generating a random population of 'N' individuals which represent the first generation. Each individual has a set of characteristics, which are the PV and battery capacity in Wp and Wh, respectively. For example, for an The final step of initialization also includes evaluating the fitness. If the convergence criteria are met, the optimization process stops. Otherwise, the optimization process repeats from Step 2 until convergence is reached. In Table 3, the parameters and convergence criteria used in this study are shown.
It must be noted that the a multi-objective optimization need not lead to unique optimization solutions. Instead, a Pareto set of solutions is usually obtained, which (in terms of GA) are the individuals with fitness functions that are non-dominated by any other individual in the search space. Here, dominance refers to the attribute of an individual to score lower (better) than all other individuals for that particular fitness function. For e.g., an individual x dominates y in the population when [31]: < f x f y i ( ) ( ) for at least 1 i i

Dependence of SHS parameters on size
As described in Section 3.7, there is a complex interplay of the various SHS parameters with respect to the SHS size. In this section, the impact of sizing on each of the system metrics (LLP, R dump , and lifetime) is presented. Fig. 7 presents a contour plot depicting the variations in LLP depending on the PV and battery size for a tier 3 load profile with VRLA battery. It can be seen that the same LLP can be achieved by either increasing the battery size or the PV size. Moreover, minimal sizing can be achieved if the knee point of the curve is chosen for a given LLP. However, the knee point, or the least distance of the curve from the origin, is a least-cost operating point only if the PV and battery are equally cheap or expensive.

LLP
In other words, for the same technology costs ($/Wp and $/Wh), the least-cost operating point is the knee point. Deviations from the knee point to achieve least-cost sizing is possible if the ratio between the technology costs is precisely known. Additionally, also shown in Fig. 7 are the battery sizes based on the days of autonomy and nights of autonomy, represented as vertical lines. It can be seen how a "rule of thumb" of 2 DOA using a "back of the envelope" method to arrive at battery size (refer to Eq. (4)) can result in oversizing of the battery, as even for 1 DOA, Eq. (4) yields a battery size of 1363 Wh for a VRLA battery assuming an 80% DOD. Using NOA method need not be very precise either, and also depends largely on the proportion of the load profile that is in the non-sunlight hours. The exact LLP achieved using such methods will also depend on the operational PV module size. In general, these "rule of thumb"-based approaches can lead to inaccurate battery sizing leading to either an expensive or an inadequately reliable system.    Fig. 8 presents a contour plot depicting the variations in R dump depending on the PV and battery size for a tier 3 load profile with VRLA battery. As seen in the figure, the R dump increases with increasing PV size. To an extent the increase in battery size helps in reducing the dump ratio. However, after around 900 Wh, the increase in battery is almost irrelevant for the considered PV and battery size range. It can already be seen that an interesting trade-off emerges between LLP and R dump . For example, to maintain an LLP of 0.05 and below (Fig. 7), almost 300 Wp of PV module is necessary. On the other hand, a 300 Wp of PV module size will guarantee an R dump of at least 0.3 as seen in Fig. 8.

Battery lifetime
The trade-off between LLP and R dump becomes more intricate when the battery lifetime is considered into the mix. Fig. 9 presents a contour plot depicting the variations in battery lifetime (L) depending on the PV and battery size for a tier 3 load profile with VRLA battery.
For the same PV size, the battery lifetime increases with increasing battery size. For the same battery size, the lowest PV size leads to relatively lowest battery activity and therefore highest lifetime. As the PV size increases for the same battery size, the battery lifetime initially decreases due to increasing battery activity contributing to higher cyclic ageing. However, at the higher end of the PV sizing range, the battery is being operated largely at relatively lower DOD levels, leading to relatively higher lifetimes than the medium PV range.

Multi-objective optimization for SHS sizing
Based on the methodology described in Section 3.7, a multi-objective optimization was performed for the various objectives. Consequently, different Pareto fronts are obtained that give the optimal PV and battery sizes that dominate at least one objective function without being worse off in the other objectives, as shown in Eq. (11). Fig. 10 shows for the tier 3 case the Pareto set of points for LLP, R dump , which also perform optimally with respect to the other objectives of lifetime and battery size. As expected, the solution space is bounded by the constraints specified in Section 3.7.3.
Figs. 11 and 12 depict the same Pareto set of points for the system metrics of LLP and lifetime, and lifetime and R dump , respectively. Some of the points that were too close to each other have been removed for clarity. It can be seen that the various system sizes represented by the points in the Pareto set perform differently across the various system metrics. As long as the different objectives are not weighted, the applied methodology does not favour a particular system size over another. Therefore, the task of optimal SHS sizing for a particular tier will   depend on selecting an optimal size based on additional filters and categorization, as described below in Section 4.2.1.
It should be noted that the GA-based MOO chooses the Pareto front based on the performance across all 4 objective functions and not just the 2 functions described in these figures. Therefore, each figure shows the Pareto points that satisfy not only the two objective functions represented on the x and y-axes, but all four objective functions stated in Section 3.7.2. This is illustrated with the help of 2 labeled Pareto points A and B in Figs. 10-12. Point A satisfies a very high battery lifetime as well as a low R dump (11.93 years and 0.1 respectively), but shows a relatively high LLP (0.07). Furthermore, the battery size, which is also an objective to be minimized in the optimization, is not explicitly represented in the Pareto fronts in Figs. 10-12, but is playing an equally important role in the selection of optimal Pareto points. For example, point A leads to a storage size of 1800 Wh for tier 3 SHS. On the other hand, point B, which seems to be specifically non-dominant for the LLP, Lifetime and R dump simultaneously (0.095, 5.81, and 0.81 respectively), actually has the lowest battery size (720 Wh), and is therefore part of the Pareto front.

Optimal SHS sizes for tiers of the MTF
Similar to the results shown in Figs. 10-12, the MOO was performed for all the tiers of the MTF. To determine the optimal system sizes per tier, a selection has to be made from the Pareto set of points allowing relative trade-offs. Firstly, three different categories of LLP are defined, viz., (a) LLP 0.1, (b) LLP 0.05, and (c) LLP 0.02. Secondly, the smallest battery size that satisfies the above LLP criteria in each category are chosen. This gives rise to unique PV-battery size combinations per tier. Accordingly, the final optimal SHS sizes for every tier of the MTF is shown in Table 4. The table also shows the system metrics for the particular PV and battery size along with the converter sizes.
Based on the methodology followed in Section 3.6, the converter sizes follow the PV rating and load profile. The PV and battery sizes, however, are a direct result of selection of a particular PV-battery combination from the Pareto set. Based on the LLP threshold, the selection of the lowest battery size to meet the LLP criteria gives an interesting mix of system sizes. For instance, as seen in Table 4, the LLP 5% criterion sees lower PV and higher battery sizes as compared to LLP 10% for tiers 3 to 5. This leads to relatively lower R dump values.

The limits of standalone SHS
As seen in Table 4, the PV and battery sizes increase drastically between the tiers, especially when going towards tiers 4 and 5. Additionally, it can be seen that no optimal solutions exist with the given constraints for tier 5 for LLP 2%. This is because larger-sized SHS that could potentially satisfy the LLP criterion would still end up compromising the dump criterion while also leading to extremely high battery sizes. This clearly shows that there is a limit to the level of electricity access a standalone SHS can provide.
For tiers 1 to 3, a small increment in system size (due to climbing up the tiers for e.g.) can be easily achieved, especially in a way such that low LLP values are guaranteed. However, tiers 4 and 5 require significant increase in system sizing, and without the kind of reliability (low LLP) achievable at the lower tiers. Additionally, the presence of high power loads also increases the required converter ratings in tiers 4 and 5. If the LLP threshold is further lowered to 1%, even tier 4 has no optimal solution based on the Pareto points. The lowest achievable LLP for tier 4 within the given constraints is 1.5%.

State-of-the-art SHS
It should be noted that the concept of LLP as a vital parameter in system sizing is usually not followed in state-of-the-art SHS. For example, present-day SHS capable of powering up to tier 2 level loads, are sized from 50 Wp, 300 Wh (for an SHS operational in East Africa) to 100 Wp, 1 kWh (for an SHS operational in Cambodia). While this may lead to suboptimal use of the SHS components of PV and battery, having a sizing approach without a reasonable approximation of LLP targets for standalone SHS catering to higher tiers could amplify the suboptimal usage.
Additionally, as most deployed state-of-the-art SHS are rated less than 200 Wp, there is a long way to go if SHS are to enable higher tiers of electrification as ubiquitously as they have been effective with tiers 1 (as pico-solar products) and 2. While tier 3 level electrification still seems to be within reach of state-of-the-art SHS, tier 4 and tier 5 level electrification demand a much bigger expansion from the standalone SHS, which may not be practical to implement when additional aspects like financial viability are considered in off-grid contexts.

Another approach to climbing up the electrification ladder
In the absence of a central grid connection in many of these off-grid regions where SHS are currently deployed, the best way to achieve tier 4 or tier 5 level of electricity access is via a microgrid. However, a centralized islanded microgrid with central PV and storage requires high CAPEX investments, which is also why standalone SHS have seen far greater proliferation than centralized rural microgrids. Moreover, the level of electrification is often a dynamic requirement, as the electricity needs keep increasing with time [3,25]. For e.g., households might not need a direct tier 4 or tier 5 connection, but might move from tier 3 to tier 4 over a span of time. In such a case, it is more practical to envisage a bottom-up SHS-based microgrid borne out of the interconnection of standalone SHS. This would enable the sharing of excess energy between the households while improving the overall system metrics of LLP and R dump . Additionally, individual battery storage requirement might also significantly reduce, along with the converter sizing, depending on the topology of the microgrid. This would especially help in alleviating the demanding system sizes at tiers 4 and 5.

Recommendations and future work
The set up of such an interconnected system is a complex research topic in itself and future work is recommended on it to examine the impact on the overall system metrics of the microgrid as compared to standalone SHS. An interconnected SHS-based microgrid model must be built to further explore the concept of climbing up the energy ladder (higher tiers of the MTF). The gains in terms of system sizing in going from standalone to interconnected SHS must be explored and quantified. Moreover, suitable energy sharing mechanisms need to be identified. These efforts are currently underway.

Conclusion
This study detailed an extensive methodology for the optimal sizing of SHS for every tier of the multi-tier framework for household electricity access. The genetic algorithm-based multi-objective optimization performed gave insights on the delicate interdependencies of the various system metrics on the SHS sizing. Moreover, meeting the energy demand of higher tiers, especially tier 5 is shown to be untenable with purely standalone SHS. Optimal system sizes for each tier of the MTF are presented, and the implications of these system sizes are discussed from the perspective of state-of-the-art SHS. Finally, an SHS interconnection-based microgrid is proposed as a potential means to climb up the so-called electrification ladder, especially to easily enable tier 4 and tier 5 levels of electricity access. The work presented in this paper is expected to shed light on the complex, multi-dimensional issue of electrification from the point of view of technical system design while exploring the intricate interdependencies of SHS parameters.

Table 4
Optimal SHS sizes that satisfy the different LLP criteria, along with the required converter sizes and the performance of these system sizes on the system metrics.