Sunlight shines on solar PV modules and generate electricity through photovoltaic effect [20] [21] . Charge controllers are connected across solar PV modules to prevent overcharging, overvoltage and quick discharge of battery [22] [23] . Generated electricity is stored in battery and some DC loads can be powered from this electricity in battery [24] [25] [26] . Electricity in battery also passes through inverter and is converted into AC electricity to be used by AC loads [27] [28] . Solar energy is used in solar lanterns, solar home lighting and solar water pumping systems [29] . Solar power systems connection technology enables them work as: 1) Stand-alone solar power system supplying only DC loads, 2) Stand-alone solar power system with electronic control circuit to extend its duty cycle, 3) Stand-alone solar power system supplying DC loads, electronic control circuit for extending duty cycle and simultaneously charging storage battery for power supply at night or whenever there is no sunlight, and 4) Stand-alone solar power system with DC/AC converter, which supply AC loads, supplies electronic control circuit for duty cycle extension and simultaneously charging storage battery for emergency supply. Therefore, stand-alone solar systems are mostly installed in remote areas without utility grid access. Batteries and other equipment needs, make them more expensive than grid-tied solar power systems, because of more maintenance and battery replacement costs after 7 - 10 years [7] . Advantageously, stand-alone solar power owners pay no monthly electricity bills because the equipment completely belongs to them after defraying installation costs [30] [31] .

DC electricity generated by solar modules passes through grid-tied inverter and is converted to AC electricity, which can be used by home electric appliances or directly fed into utility grid through a net-meter or a bi-directional meter [7] . Grid-tied solar power systems are cheaper and simpler than stand-alone solar systems because they do not need batteries and other expensive equipment, since the utility grid is the virtual battery bank [14] [32] [33] [34] . Grid-tied solar power systems are more cost effective and accepted, easy to operate with low maintenance, can be sized to meet any scale of power requirements, nor sized to meet peak loads demands as the excess electricity is fed into utility grid, and drawn from it, whenever, it is needed [7] . Grid-tied solar power owners save money using net meters, because they sell electricity to utility grid whenever their solar system produces more electricity than what their electrical appliances can consume. Conversely, utility grid supplies electricity to grid-tied solar power systems owners, whenever there was a deficit from solar power systems [35] . Mainly because the grid-tied solar power systems have no batteries, whenever there is no sunlight and utility grid is down, they would have no electricity, especially at nights [7] [31] [36] [37] [38] .

Farhat and Reichelstein [45] used levelised cost and profitability of polygeneration to align both fuels and generated products on same per unit bases to reliably compare polygeneration with single-electricity generation using natural gas or wind alternatives. Net present value (NPV), profit index (PI), payout time, and cost of energy were used to evaluate energy economics, but none of these methods has produced consistency in comparison to cost-benefit. Furthermore, the cost of energy (($/kWh) approach has the following drawbacks: 1) It may not produce electricity as its end product, 2) It is difficult to obtain equivalent cost of non-electricity products in electricity cost quantities, and 3) Underestimation of actual cost of electricity [45] .

Therefore, assessment of technological and economic effectiveness should combine operational valuation with economic trade-offs for economic gains. Hence, investment planning, trade-offs in operational flexibility, techno-economic analyses using input-output comparison coupled with process configuration, should be embraced for stand-alone and grid-tied solar PV systems [45] . Flexibility limits were imposed on optimized operations to determine operational constraints of solar electricity generation based on economics. These helped to determine profit margins and real options for valuation.

Also, breakeven calculation is the ratio of lifetime costs to lifetime solar electricity plant production [45] . The algorithms development for lifetime costs and lifetime stream of benefits for solar electricity plant production, follow.

The cost of solar electricity (PV) consists of capacity per unit output C_e, time averaged fixed operating cost per unit output f_e, and time averaged variable cost per unit of output v_e. So the cost of solar electricity generated from PV is:

COEPV = c e + f e + v e (1)

The cost of capacity per unit kilowatt of solar electricity is:

c e ( $ / kW e ) = P e n ⋅ C F ⋅ ∑ j = 1 Y a j δ j (2)

where P_e is price of unit capacity to deliver one kilowatt of solar electricity per hour. The components of this cost include procurement and construction, import duties, and other ancillary services. The stream of benefits yield of solar electricity from PV system is over Y years, with n ⋅ a j ⋅ C F kilowatts for year j, n = 8760 hours in any one year, a_j is derating failure factor (ageing and solar technology dependent efficiency loss factor), CF is capacity factor (fraction, because PV systems plant is not operated 100% at all times), δ is the discount factor, which is a decimal value multiplied against cash flow to discount it back to today’s value (present value) [45] [46] . Furthermore, discount factor: 1) increases over time, as its decimal value becomes smaller, 2) provides explicit factors for each time period, to enable us see the effect of compounding more vividly, 3) makes discounted cash flow (DCF) model easier to audit and 4) is alternative to using either NPV or internal rate of return (IRR). And ξ is the discount rate, which converts one-time costs into annualized costs [46] .

The discount factor is [46] : Discount   factor = 1 / ( 1 × ( 1 + Discount   rate ) Periodnumber )

Accordingly, using the breakeven formula, a convenient discount factor for solar electricity projects comparison is [45] : δ = ( 1 / 1 + ξ ) 2 . Furthermore, fixed operating costs change yearly, and average fixed operating cost for producing one kilowatt of PV solar electricity incurred per year is f e j ($/kW_e). Fixed operating costs include, labor, administrative overheads, maintenance and insurance [45] . Costs are assumed proportional to installed capacity, and independent of actual solar electricity produced:

f e = ∑ j = 1 Y f e j δ j n ⋅ C F ∑ j = 1 Y a j δ j (3)

Conversely, variable costs change over time as ancillary loads, foreign exchange conversion expenses and interest charges [45] . Let v e j ( t ) ($/kW_e), be time-dependent variable cost of one kilowatt of solar electricity in year j and, v e j ($/kW_e), yearly average variable cost, then yearly averaged variable cost becomes:

v e j = 1 n ∫ 0 n v e j ( t ) d t (4)

Consequently, over project lifecycle, average time variable cost for 1 kW of solar electricity:

v e = ∑ j = 1 Y v e j n ⋅ C F ⋅ a j δ j n ⋅ C F ∑ j = 1 Y a j δ j = ∑ j = 1 Y v e j a j δ j ∑ j = 1 Y a j δ j (5)

Similarly, P e j ( t ) , P e j , and P e are respectively, time-dependent solar electricity price for year j, yearly average price of solar electricity in year j and time averaged price for solar electricity. From breakeven analyses definition for COEPV, PV solar electricity production plant under investigation is cost-effective if and only if [45] :

P e > COEPV (6)

Therefore, cost-effectiveness is measured by that quality of PV solar electricity plant, which sustains positive net present value (NPV), and equates with breakeven analyses.

Frequentist designs focus on planning experiments through appropriate choices of sample sizes, whose long-run average actual accuracy equals long-run average reported accuracy [47] . The 95% classical confidence interval of unknown success probability θ, becomes Jeffreys equal-tailed 95% confidence interval:

C J ( x ) = ( q 0.025 ( x ) , q 0.975 ( x ) ) , (7)

where q α ( x ) is αth-quantile of Beta ( x + 0.5 , 20.5 − x ) distribution. Applying the modified Jeffreys equal-tailed confidence interval for large m, and using local smoothing kernel over range θ ± 2 σ supplies Beta ( a ( θ ) , a ( 1 − θ ) ) distribution K σ , θ ( . ) , where the frequentist coverage is 1 − α ( θ ) , and leads to σ-local average coverage:

1 − α σ ( θ ) = ∫ 0 1 [ 1 − α ( θ ) ] K σ , θ ( θ ) d θ = ∑ x = 0 n ( n k ) [ Γ ( a ( θ ) + a ( 1 − θ ) ) ⋅ Γ ( a ( θ ) + x ) ⋅ Γ ( a ( 1 − θ ) + n − x ) ]     ⋅ [ Γ ( a ( θ ) ) Γ ( a ( 1 − θ ) ) Γ ( a ( θ ) + a ( 1 − θ ) + n ) ] − 1     ⋅ ∫ C J ∗ Beta ( ( θ ) | a ( θ ) , a ( 1 − θ ) ) d θ (8)

Equation (8) is beta-binomial predictive distribution, where Beta ( θ | a ( θ ) , a ( 1 − θ ) ) is beta density function having specified parameters. Also, π ( θ ) ∝ θ − 1 / 2 ( 1 − θ ) − 1 / 2 for θ. These intervals consist of upper and lower α/2-quantiles of posterior distribution for θ. Furthermore, for two-sided test of null hypothesis, θ ∗ = | θ ∗ ∗ | , is absolute value of real parameter θ ∗ ∗ , so that θ ¨ ∗ = [ 0 , ∞ ) . But, p ( x ) = S ( 0 ; x ) is a two-tailed p-value for hypothesis testing, such that: θ ∗ ∗ = 0 . Also, P ¨ ∗ ( θ ¨ ∗ ≤ 0 ) = S ( 0 ; x ) , and

P ¨ ∗ ( θ ¨ ∗ ≤ 0 ) = P ¨ ∗ ( θ ¨ ∗ = 0 ) (9)

p ( x ) = P ¨ ∗ ( θ ¨ ∗ = 0 ) , is p-value, which is probability that null hypothesis is true [49] [67] . τ ¨ = θ ¨ ∗ → θ , is defined by

τ ¨ ( θ ¨ ∗ ) = 1 ( 0 , ∞ ) ( θ ¨ ∗ ) (10)

That means, θ ¨ = 0 , if θ ¨ ∗ = 0 . Also, θ ¨ = 1 , if θ ¨ ∗ > 0 . Therefore, p ( x ) = P ¨ ∗ ( θ ¨ ∗ = 0 ) , which mandates, P ¨ ( θ ¨ = 0 ) = p ( x ) , and by implication:

P ¨ ( θ ¨ = 1 ) = 1 − p ( x ) (11)

For normally distributed noise ε ~ N ( 0 , σ 2 ) , the likelihood function is [48] :

L ( y | θ ) = ∏ k = 1 m ∏ l = 1 d k 1 2 π σ k l 2 exp ( − 1 2 ( y k l − y k ( t l , θ ) σ k l ) 2 ) (12)

where m model outputs y k ( t l , θ ) and d k data instants, at such time points t l . The maximum of L ( y | θ ) , which is best model fit to data, gives point estimate θ ^ of parameters [48] . The threshold Δ α = Q ( χ d . f 2 , α ) , is α quantile of χ²-distribution with degrees of freedom (d.f), and confidence intervals determined through profiles:

{ θ i | − 2 log ( P L ( y | θ i ) | L ( y | θ ^ ) ) < Δ α } (13)

While penalisation removes difficulties in stability of parameter estimates, shrinks semi-Bayes or partial Bayes estimation without satisfying Bayesian requirements, it allows us tolerate degrees of bias as a trade-off for parameter variability estimates [49] . Therefore, the Penalised log-likelihood becomes [49] :

ln { L ( β ; y ) } − r ( β − m ) 2 / 2 (14)

where, r/2 is penalty weight factor relative to original log-likelihood, m is bases for good value information and β are unknown parameters. However, every statistically significant result is subject to a type-I error, of falsely rejecting the null hypothesis as no bias in point estimates. In multilevel analyses with a few clusters (as in this study), [50] assert that frequentist methods provide unbiased estimates and accurate inference. Thus, results are more precise as they occupy narrower bands, while parameter bias estimate magnitudes shrank towards zero.

While posterior distribution is chosen by nature for priors, statisticians choose parametric distributions to maximise minimum Kullback-Liebler information [51] , acquired over a confidence distribution. Bayesian methodology is used when prior distribution is determined from relative frequencies, and Frequentist approaches are used without knowledge of such parameter distributions. The condition for total ignorance of prior distribution has a lower bound of 0, and p-value is used in place of any posterior probability. Whenever prior distribution is completely known, posterior is used for inference, representing a higher bound of 1 [47] [51] [52] which is a convex set (0, 1 problem). From hindsight, convex functions are always minimising and lead to optimal solutions in the interval of convexity, with one minimum [52] [53] . The analyses follow.

We define H as the Hessian, H_r as the relaxed Hessian, L as the Lagrangian and L_r is the relaxed Lagrangian (it is the largest functional below L). Assuming the minimum cost occurs when δ ≤ 1 . It means: 1) Minimum cost E' cannot be less than the project cost E, because H r ≤ H as L r ≤ L for each non-negative λ and μ, 2) H_r is polyconvex as its functional is minimum, 3) The constrained minimum cost E' equals E. Furthermore, the minimising functions represent weak oscillatory limits for L. But, H_r is polyconvex and H r ≤ H [52] [53] . The proof of polyconvexity using convex functions shows that multilinear functions, which envelope is H_r is actually below H, and it is also minimising. Therefore, [52] :

d ( ρ ) = { 2 ρ ,     0 ≤ ρ ≤ 1 1 + ρ 2 ,     ρ ≥ 1 (15)

The square root of ρ is a convex function and d ( ρ ) is also convex and increasing. The composition of d ( ρ ) is also convex. The linear Jacobian terms ∓ 2 J , leave H_± convex as functions with extra arguments. But, H_r is maximum of the two functions, H_± and whenever J is identified with det [ ∇ u ∇ v ] , H_r must, therefore, be polyconvex [52] . Also, d ( ρ ) = 1 + ρ 2 and for large ρ, the H_± functions become: 1 + | ∇ u | 2 + | ∇ v | 2 . For small ρ, the comparison between H₊ and H₋ depends on:

2 ( m + n ) 1 / 2 − n ≥ 2 ( m − n ) 1 / 2 + n (16)

The above inequality holds for: m ≥ n ≥ 0 and, m + n ≤ 1 . So, m = | ∇ u | 2 + | ∇ v | 2 and n = 2 | J | . The maximising choice of sign is that for which ± det [ ∇ u ∇ v ] equals | J | . The argument ρ is exactly γ in the definition of H_r, where max H_± coincides with H_r [52] [53] . However, H_r is below and less than H mainly because 2ρ is below and less than 1 + ρ 2 . The difference: ( 1 − ρ ) 2 = ( 1 − γ ) 2 in the range 0 < γ < 1 , is the saving in investment cost that was achieved through homogenisation and convexification of the solar PV system.

We designed PV systems choosing the type of photovoltaic module and some system parameters. Znshine PV-Tech ZX250 (48) MS module [54] was used having the following specifications on nameplate: V r m = V m p = 48 . 7   V , I r m = I m p = 5.14   A , V o c = 59.6   V , I s c = 5.49   A and each PV module costs US$375.00 (Table 2). Efficiency was estimated to be 0.85, 0.9 and 0.9 respectively for batteries, charge controllers and inverters. The system voltage was set to 24 V_dc and 5 hours was the average peak sun-hours per day in Rwanda [3] .

The MK 8A8DLTP-DEKA 12V 245Ah AGM Battery, able to support PV array was selected with parameters: C_b = 245 Ah, V_b = 12 V, D_disch = 80%, B_cost = US$588.00, D_autonomy = 2 days [55] (Table 3).

The Morningstar TS-MPPT-60 Tristar Mppt 60A charge controller was selected, with specifications: I_cc = 60A, C_cost = US$598, safe factor (F_safe) = 1.25 [56] (Table 4).

During inverter sizing and design, power ratings of appliances that may run simultaneously were added as follow: P = (96 + 210 + 75 + 15 + 1100 + 250 + 15 + 25 + 150) W = 1936 W (non-inductive loads) and appliances with large surge

currents including motors were calculated as: 245 × 3 = 735 W (inductive loads) (Table 5 and Table 6). 1.25 scale factor was used to multiply both inductive and non-inductive loads to allow for miscellaneous and possible system expansion [57] . The handling power for inverter becomes: (1936 + 735) W × 1.25 = 3338.75 W. The NIMTEK MM4000 Pure Sine Wave Off-grid Inverter was selected, a Solar Inverter with 4000 Watts, 24 Volt DC to 220 Volt AC, costing US$660.00 [58] .

The house is located in a residential area where electricity tariff is US$ 0.23/kWh. Then:

Annual savings = 10.463 kWh/day × US$0.23/kWh × 365 = US$ 878.36 saving per year

Payback period = US$15932.07/863.325US$ = 18.13 years.

The total energy requirement for typical residential house case study was 7204 Wh/day. Taking energy loss factor in solar system to be 1.3 [59] , Wh/day needed from PV modules = (7204 × 1.3) Wh/day = 9365.2 Wh/day. Thus, for PV panel sizing, peak sun hours of 5 hours [3] , direct system kilowatt size at STC = 9365.2/5 =1873.04 Wp. Using Znshine PV-Tech ZX250 (48) MS module [54] , number of modules required = 1873.04/250 = 7.49 (approximately 8) modules. The total cost of modules, is: 8 × US$375 = US$3000.00. Since loads are AC and assuming power factor of all appliances vary between 0.85 and 0.99. Using, 1.18 multiplier to convert 1873.04 Wp into VA [59] : VA output = 1.18 × 1873.04 Watts = 2210.18 VA. The cost of grid-tied inverter which is able to support 2210.18 VA is US$210 [61] . The cost of PV modules and inverter is: US$3000.00 + US$210.00 = US$ 3210.00. Assuming 20% of total cost as price of cables, net metering devices, labour, design and control devices, 20% of PV system total cost will be US$ 3210 × 0.2 = US$642 [62] . Thus, overall cost for grid-tied solar power system for typical house case study in Rwanda was US$ (3210 + 642) = US$3852.00. This overall cost was regarded as Freight on Board cost (FOB) before shipment to Rwanda.

Assume Carriage and Freight (C & F) cost to be 10 % of FOB = 0.1 × US$3852.00 = US$385.20.

Contingency (Assume 5% of Sub-total for (FOB + C& F)) = 0.5 × US$ 4237.20 = US$ 211.86

Grand Total (FOB + C& F + Contingency) = US$ (385.20 + 211.86 + 3852) = US$ 4449.06

Annual savings = 365 × US$ 0.23 × 9.3652 = US$ 786.20

Then payback period will be equal to US$ 4449.06/US$ 786.20 = 5.65 years

According to [59] , some drawbacks commonly associated with stand-alone PV systems include: 1) Batteries waste between 30% and 40% of costly solar power to charge inefficiencies, 2) Off-grid users waste or throw away surplus summer power, which is usually more than 50% of their requirements, 3) Batteries are unable to store more than 50% summer extra power for use in winter, when less than 50% solar requirements would be available, 4) During winter, lifestyle changes and adjustments to use less than 50% normal summer solar average, are unavoidable, 5) Planning for battery maintenance throughout PV system’s life, 6) Planning for batteries bank replacement every five (5) or so years, throughout home occupiers lives, 7) PV arrays must face specific direction (due East or West or South) at correct angle (This is not so with Grid-Tied), and 8) Need for back-up generator for long stretches of cloudy days, coupled with adjusting to aggressively increasing gas prices.

Whereas our analyses show grid connected PV solar system is the preferred choice, [39] and [59] , suggest this preference from, using rule-of-thumb, which goes against tradition and convention in solar power usage. Furthermore, there are hardly any real requirements at majority of latitudes about exact roof facing. Also, consideration to avoid Trees and Shading [60] , have been shown to have more impact than roof facing in Grid-Tied PV systems. In addition, East-West facing house roofs produce more Grid-Tied power than those facing the South [39] .

Maheshwari and Bruninga [34] and [54] suggest producing excess 1 kW in summer feedback into the grid goes to feed nearest 1 kW load (neighbor). The power utility charges that neighbor and utility does not need to generate 2 kW worth of electricity, leading to avoided generation. Consequently, this translates to saving more than 4 kW energy of coal equivalent electricity for later use. In addition, utility does not need peaking units to meet peak demand, especially at normal prices. The economic benefits of Grid-Tied PV system include [34] [59] [63] [64] : 1) Homeowner wins by pushing meter 1 kW backwards, 2) Homeowner neighbour also wins by receiving 1 kW at standard electricity rate, 3) Power utility wins by avoiding costs of not burning 4 kW of equivalent coal or other fossil fuel units, 4) Utility does not need to buy 1 kW of electricity at exorbitant rates from neighbouring or adjacent utility, 5) Homeowner receives Solar Renewable Energy Credits (SRECs) for PV systems, 6) Environment also wins, from deferred burning of coal or other fossil fuels.

The Rwandan REFITs are subjected to be reviewed by Rwanda Utilities Regulatory Authority (RURA) and other energy policy makers. REFITs policy for renewable energies, like solar, wind and geothermal are already in use but have not been issued and considered for Grid-Tied PV system [1] [65] . The motivations to install PV REFITs include: 1) Saving or earning money, 2) Increasing household energy independence, 3) Protection against higher future electricity costs, 4) Desire to improve environment, 5) Showing environmental commitment, and 6) Higher disposable income. Conversely, barriers to installation include: a) Lower disposable income, b) Age of customers, c) Likelihood of moving house within 5 years, d) Probability of losing money as they move house, e) Difficulty of obtaining reliable information, f) Aesthetics and neighbor approval, g) Uncertainty of potential system performance [66]