# The cost of reliability in decentralized solar power systems in sub-Saharan Africa

## Abstract

Although there is consensus that both grid extensions and decentralized projects are necessary to approach universal electricity access, existing electrification planning models that assess the costs of decentralized solar energy systems do not include metrics of reliability or quantify the impact of reliability on costs. We focus on stand-alone household solar systems with battery storage in sub-Saharan Africa using the fraction of demand served to measure reliability, and develop a multistep optimization to compute efficiently the least-cost system with the fraction of demand served as a design constraint, and take into account the daily variation in solar resources and costs of solar and storage. We show that the cost of energy is minimized at approximately a 90% fraction of demand served, that current costs increase, on average, by US$0.11 kWh–1 for each additional ‘9’ of reliability, and that this reliability premium could be as low as US$0.03 kWh–1 in a plausible future price scenario.

## Main

The United Nations Sustainable Development Goal no. 7 describes a major global task: ‘‘Ensure access to affordable, reliable, sustainable, and modern energy for all’’1. In 2017, 1.06 billion people remain without access, and more than half of them live in sub-Saharan Africa (SSA), the geographic focus of this analysis2. Historically, electrification from centralized national and regional electric grids was the only path; now, interest and investment in decentralized, stand-alone options from solar home systems (SHS) to community minigrids is growing3. Private capital investment in SHS providers increased from US$3 million in 2012 to 381 million in 2015–2016, and the deployment of solar energy products for electrification grew similarly4. We use ‘decentralized’ to describe systems that are self-sufficient and independent of a connection to a larger grid. This definition applies to a range of applications, but here we focus on isolated, household-scale systems. Multiple studies point to the efficacy of solar–diesel–battery hybrid systems that use diesel generators to increase reliability5,6; however, recent work that explored future low-carbon energy systems points to an increasing reliance on solar–battery systems7 and high penetrations of SHS8, which provides motivation to study SHS at high levels of reliability. The centralized and decentralized paradigms have advantages and disadvantages. Centralized grids in SSA utilize economies of scale3 to achieve lower costs of energy, but often do not reach rural areas. Even where the grid is present, some households are left ‘under the grid’ because of high connection costs and long wait times for connections9. When the existing grid service is unreliable, some consumers use SHS as a back up10. Decentralized systems can be flexibly and rapidly deployed to meet basic energy needs in many geographies and communities, but high costs and the challenges of a variable resource raise questions about whether they are a viable long-term economical solution for electrification. Nonetheless, the International Energy Agency predicts that 60% of the connections needed for universal electricity access will come from decentralized systems11, but the eventual outcome will be influenced by the cost trends, policy and regional electrification plans that guide investment. These electrification plans rely on cost minimization models that compare grid expansions to decentralized options12,13,14,15, which are most frequently solar photovoltaics (PV) with battery storage. Yet, we currently have a limited understanding of the economics of decentralized systems and their reliability across large spatial scales. Many models assume constant per-unit costs of energy for classes of decentralized systems12,13,14,16,17, whereas others scale the system size by location-specific solar resources, but assume constant storage-to-solar capacity ratios18,19 or a single reliability level20,21. We are aware of one analysis that investigates the cost of reliability for decentralized systems, but the results are isolated to one location22. Ultimately, the centralized versus decentralized debate needs to be answered within the context of least-cost electricity planning tools. Although such tools have been applied in the African context12,13,14,15,16,17,18,19,20,23,24,25,26, their computational complexity and the lack of available knowledge of decentralized system costs precluded the integration of decentralized pathways into the models. In this study, we investigated how the design of decentralized systems for different levels of reliability affects their cost, how the costs of decentralized solar systems compare with those of national grids when designed for an equivalent reliability and how changing commodity prices can affect these relationships. We found evidence that decentralized solar–battery systems are on the cusp of reaching ‘grid parity’ in both economic and reliability terms in many parts of SSA. Furthermore, using aggressive forecasted technology costs taken from the literature, we show that, in the future, a large fraction of the continent could be served by decentralized systems with better economics and reliability than the existing grid. We propose a method to identify the optimal PV and battery capacities for decentralized systems using 11 years of location-specific daily solar resource data (at a 1° latitude and longitude resolution) and across a large range of reliabilities (measured in terms of the fraction of demand served (FDS)). We also rigorously quantified the cost to improve the reliability at each location and show that this ‘reliability premium’ has a strong spatial variability, and increases the levelized cost of energy (LCOE) by US$0.05–0.15 kWh–1 for each order of magnitude improvement in reliability (for example from 99–99.9%). These results point to the potential for decentralized solar systems to provide a very high reliability service at costs that are competitive with existing, often highly unreliable, grid infrastructures. The method we employed is computationally scalable and can be used for the rapid analysis of different cost assumptions (an open-source implementation is available at https://emac.berkeley.edu/reliability). The model can also be integrated into planning models to capture the trade-offs between a spectrum of technology pathways that range from centralized grid options to fully decentralized systems. We compared our method to existing approximations, and found that simpler methods provide a good, although not perfect, estimate at a FDS below 99%, but that these methods are inadequate to quantify LCOE variation at a higher FDS.

## A framework for the cost of reliability

To quantify reliability, we define and use the FDS, which is, over the analysis period, the sum of all the energy delivered divided by the sum of all the energy demanded. An FDS of 1 indicates perfect reliability. The FDS is similar to the more common Average Service Availability Index (ASAI)27, which describes the fraction of time that the service is available, and can be estimated by the World Bank Enterprise Surveys for many countries, as shown in Fig. 128. FDS is effectively the ASAI weighted by demand—ASAI does not distinguish between outages when demand is high versus low, but outages during the periods of high demand will have a greater impact on the FDS. Note also that the FDS = 1 – ESP, where ESP is the energy shortfall probability22. Studies that employ the optimization software HOMER20,22 use a single year of measured radiation to construct estimates of the FDS; we used 11 years of remote sensing data to increase the robustness of the estimate. Our approach could, alternatively, use ASAI or the System Average Interruption Duration Index (SAIDI)27, but we used the FDS as it relates more directly to energy demanded instead of time. We only modelled solar resource driven outages; evidence from the field suggests technical failures are much less significant than resource outages29.

We quantified the FDS because it has an important impact on the cost and utility of decentralized solar systems, and because there is significant variation in the reliability of grids in SSA, as illustrated in Fig. 1. A 2014 simulation for a microgrid in Mali showed that as the FDS approaches 100%, there is a log-linear relationship between the reliability and added cost to achieve that reliability, and that the optimal ratio of battery to photovoltaic capacity changes significantly for different target reliabilities22. The optimal battery-to-PV ratio also depends on local weather patterns and demand patterns. For example, additional battery capacity might be the best choice at a location that has frequent, but intermittent, cloud cover, but would be ineffective in regions with prolonged rainy seasons. The software HOMER uses simulation and searches over a large parameter space, which makes it adequate for single-location studies20,22 but unsuitable to evaluate costs on continental scales or for use as a subproblem in a larger optimization.

Our approach preserves the nature of the solar array versus battery bank capacity trade-off by computing an isoreliability curve (equivalent to a Pareto optimal frontier), that is, the set of all system designs that achieve a desired FDS for a particular location. With an isoreliability curve, the cost-minimizing capacity of storage and solar given their costs (battery, PV module, racking, charge controller and so on) can be found by simple line search. We constructed isoreliability curves through simulation for each location in SSA at a 1° latitudinal and longitudinal resolution and for the FDS of interest and stored the results. This approach enables modellers to then analyse a large number of cost scenarios with minimal computation, which enables detailed optimizations at a high spatial–temporal resolution and geographic scale, and so bridges the gap between detailed local models and wide area studies (Methods illustrates how to adjust economic assumptions).

An isoreliability curve depends on hourly consumption patterns; night-time load requires more storage than daytime load. However, because the curves are constructed per unit of daily load, they are independent of the average load (kilowatt hour per day). This independence enables the method’s scalability. As location-specific load-shape data are not available for the regions we studied, and because our solar insolation database provides daily (rather than hourly) data, we present our results assuming a constant load throughout the day (or, equivalently, with a load factor of 1). To test this assumption, we performed sensitivity analyses using different load curves: (1) constant load, (2) all the load is from 18:00 to 6:00 and constant during that time, (3) all the load is from 6:00 to 18:00 and constant during that time and (4) the load follows a representative profile that contains an evening peak that was empirically measured on a rural microgrid in Uganda. These tests show that the qualitative results we present here are robust to other load shapes. Specifically, we found that across the FDS, constant load yields approximately the same costs as a measured load profile from Uganda with a night-time peak, and that concentrating demand at night raises the LCOE by up to US$0.15 kWh–1, whereas concentrating demand during the day lowers the LCOE by US$0.10 kWh–1 (Supplementary Note 2). We also assume that the load is identical every day; the incorporation of stochastic load models into this framework is an open technical challenge.

## Commodity prices and the costs of solar electricity

We developed scenarios from electricity access ‘Tier 5’ defined by the World Bank’s Energy Sector Management Assistance Program (ESMAP) and computed the LCOE across SSA using the economic assumptions in Table 1. This tier is the highest level of access for the household and productive uses of electricity and includes explicit metrics for reliability, capacity and consumption30. In particular, assuming FDS and ASAI are equivalent because of a constant load, Tier 5 requires an FDS of 95% by specifying that the service is available for 159 out of 168 hours per week. Figure 2 shows that the current LCOE varies by about US$0.15 kWh–1 across SSA, and that the potential future cost reductions are greater in magnitude than the current spatial variation. Lower component costs in the future scenario reduce the average LCOE, and also lower its coefficient of variation, which shows that the cost declines have a disproportionate impact on the higher cost locations. The future scenario entails aggressive cost reductions that could plausibly be realized by about 2025, but we did not forecast the exact time frame of cost reduction (Methods). The location of high- and low-cost areas is similar to that in earlier work with simplified cost models20, but we found generally higher costs in the current scenario and that the cost reduction effects outweigh current spatial variation (Fig. 2). To understand the trends in the costs incurred to achieve a desired reliability, we show the density of LCOE across a range of reliabilities using all the locations under the current and future cost scenarios (Fig. 3). The figure indicates three important results: (1) LCOE increases logarithmically as FDS approaches one, but has a minimum slightly above a 90% reliability, (2) reducing the component costs has a disproportionate impact on reducing the premium for high levels of reliability and serves to flatten the LCOE-to-reliability curve and (3) the spatial variance in LCOE increases at a higher reliability. This logarithmic scaling, which was previously predicted by a probabilistic model that approximates the isoreliability curves31, simplifies decision-making for sizing reliability and estimating the cost of reliability. The LCOE minimum arises because there are constant fixed costs associated with the system (for example, inverters and wiring), but they are spread over fewer kilowatt hours consumed as the FDS declines. This implies that it is not economical—on an LCOE basis—to design systems with a less than 90% FDS, given these assumptions; this minimum is approximately stationary in different cost scenarios. Synthesizing Fig. 1 with Fig. 3, the current LCOE of decentralized solar varies by approximately US$0.2 kWh–1 in the range of the FDS observed on the grid.

The possible future cost scenario indicates that the reliability premium declines and has less spatial variability as component costs decline. The decline in variance in LCOE is greater than that we would expect as a statistical implication of the lower mean LCOE, so the reduction in component costs causes a disproportionate reduction in the reliability premium for high cost areas.

We found that the logarithmic growth in the LCOE as the FDS approaches one along with the existence of the reliability minimum are well-captured by the parameterized relationship for each location in equation (1) and across locations in equation (2):

$${\rm{LCOE}}_i = - a_i\frac{{\log _{10}(1 - {\rm{FDS}})}}{{{\rm{FDS}}}} + b_i\frac{1}{{{\rm{FDS}}}} + c_i$$
(1)
$${\rm{LCOE}} = - a\frac{{\log _{10}(1 - {\rm{FDS}})}}{{{\rm{FDS}}}} + b\frac{1}{{{\rm{FDS}}}} + c$$
(2)

In equation (1), i indicates a particular location sampled at a 1° latitudinal and longitudinal resolution across SSA. The parameters ai, bi and ci are calculated using a least-squares regression to each location. For high reliabilities, ai gives the reliability premium at location i: for every ‘9’ of reliability, LCOE increases by approximately ai US$kWh–1 (Fig. 4). The reliability premium is driven by the additional storage and solar capacity needed to ensure the capacity during a low solar resource (Supplementary Note 1 gives additional discussion on the optimal system size); however, we did not observe any patterns in the distribution of outages or component costs across the locations (for example, storage cost dominating in one region and solar costs dominating in another). At the current costs, most of SSA has a reliability premium of US$0.05–0.15 kWh–1 per 9 of FDS, although with high variation. The future reliability premium could be as low as US$0.03 kWh–1 in most of SSA. ## Alternative methods to estimate LCOE A motivating hypothesis of this analysis is that we can improve on methods that estimate the cost of decentralized solar using a fixed ratio of solar PV capacity to battery storage capacity. To test this, we checked how well the mean annual insolation alone predicts our calculated LCOE (Fig. 5). Mean insolation explains roughly 80% of the variation in the LCOE for a FDS of 0.9 (Fig. 5a) and further loses its predictive power relative to our model as the FDS increases. Intuitively, this loss in predictive power is because the reserve capacity to account for a temporal variation in the solar resource is more significant at a higher FDS and starts to determine costs more than mean insolation. This suggests that methods such as the one we propose are increasingly necessary at high FDS values. At a lower FDS, one could use a simple linear model driven by annual insolation. However, that model would need to be parameterized with output data from a model such as the one presented here, and it could still incur significant errors, especially at low annual insolation levels (Fig. 5). We also found that the optimal capacity of storage and solar is more variable across locations at higher FDS values, which implies that it is especially important to compute the optimal system design at higher FDS values. We calculated the cost penalty of using a suboptimal system design by first computing costs with the ratio of storage to solar capacity fixed as desired FDS changes. We then compared this cost to the optimal cost at different FDS values. The penalty incurred for the current cost scenario increases with the FDS to around US$0.10 kWh–1 in many regions, although the spatial variation is significant (Supplementary Note 1). We conclude that, although the inaccuracies of a simple model (based on mean annual insolation and an approximate storage-to-solar ratio) may be acceptable to estimate the cost of decentralized solar systems at FDS values below 99%, at a higher FDS it is necessary to use an optimization that accounts for local weather patterns on a daily scale.

Referring to Fig. 1, we see that many countries in SSA have a grid reliability of less than 99%. This is the range in which simpler estimates of the LCOE, based on constant storage-to-solar ratios, are relatively close to the estimates from our model. Our more detailed model remains important for several reasons. First, results from our detailed model are needed to validate those of simpler ones at different reliabilities, which contributes an understanding of the threshold at which it is appropriate to use one model versus another. Second, and perhaps more importantly, the model is needed in cases where a reliability higher than that of the current national grids is desired, for example in the planning of future power systems that have a reliability on par with that of the rest of the world (for example, the United States reported its 2015 ASAI as >99.9% (ref. 32).

## Decentralized solar LCOE and grid tariff comparison

We now compare the LCOE from decentralized solar to the grid using equivalent performance metrics, namely the FDS. We computed the ASAI for most countries in SSA using the World Bank Enterprise Surveys28 (Fig. 1). These surveys record the frequency and duration of outages reported by businesses, although we note that these numbers are typically 6–7 times higher than those reported by utilities33. Treating FDS and ASAI as equivalent under our constant load assumption, we computed the LCOE of a decentralized system that provides the ASAI reported for the grid. This enabled us to compare the cost differential between the decentralized solar LCOE and grid tariffs34 at an approximately equivalent quality of service (Fig. 6). Although grid tariffs are often both directly subsidized and cross-subsidized between customers, and do not accurately reflect the cost of the service17, a comparison of tariffs facilitates our understanding as to where utilities might face competition from decentralized solar and our understanding as to how decentralized solar compares to business as usual. Additionally, the grid reliability may be unacceptably low in some regions; however, adjusting for reliability provides a more appropriate comparison for countries with a reliability above 90% (Fig. 1). Grid tariffs and outage rates are reported in Supplementary Note 3.

Figure 6 shows that in some areas, particularly in West and East Africa, the costs of decentralized solar are approaching grid parity at an equivalent reliability, and, in the future, could become cheaper than current grid tariffs. A benchmark study for comparing the current cost of solar against the grid20 finds that to include the estimated cost of grid extension in addition to grid tariffs results in one-third of the population of Africa being most cost-effectively served by solar, but with the vast majority in rural areas. Our results show a current cost difference much less favourable to solar because we do not estimate a cost of grid extension; however, we project that future low costs for solar and storage could enable decentralized systems to threaten the utility business model in many countries, even in urban areas where grid extension is not necessary. Many of the countries in which decentralized solar is competitive are also those with relatively low rates of electrification, which suggests that they could be ideal locations for decentralized electricity solutions. In particular, Mali, Liberia, Uganda and Rwanda all have low electrification rates35 and a relatively low cost difference between decentralized solar and the grid. Results by country are available in Supplementary Note 4.

There are many factors not analysed here that determine the cost of grid service to the customer, including subsidy, connection fees and usage; and there are also complex factors that determine the cost for the utility to extend the service. However, this suggests that particular national utilities could face increasing competition from decentralized solar on the individual household scale, and that certain countries have grid tariff structures and solar resource characteristics that make decentralized solar a competitive option.

## Discussion

Our first major conclusion is that as solar and battery costs decline, decentralized systems with a high reliability that meet ESMAP Tier 5 criteria are likely to become cost-competitive with the grid in a large portion of SSA. This cost parity with the centralized grid is within reach both because of declining costs in solar and storage and because centralized systems have significant costs to offset losses (both technical and non-technical) and to build and maintain the transmission and distribution infrastructure. The magnitude of the cost declines is, of course, strongly dependent on the assumptions input to the model; these assumptions can be explored in an online, open-source version of the model (https://emac.berkeley.edu/reliability).

Many countries for which cost parity is likely also have low rates of electrification. Our results therefore highlight the risk that the standard aid agency pathway to fund large grid-development projects (which have long lead times and complex regulatory process) could be stymied by private decisions to build decentralized systems (whose lead times are extremely low and require little to no government involvement).

Low reliability premiums present an additional challenge to the centralized grid paradigm; whereas a centralized grid reliability is outside the customers’ control, in decentralized systems customers can influence their reliability with both upfront decisions and real-time curtailment decisions. For example, in many parts of our study region, a customer considering a system with 7,900 hours of service per year (90% reliability) could add another 790 hours of service (to 99% reliability) for less than US$0.10 kWh–1. This cost could lower to US$0.03 kWh–1 if aggressive cost declines in PV batteries are realized. From a planning perspective, it would be useful to compare these reliability premiums for the grid to direct investment; however, the premium for the grid is not well understood because of its complexity. This is an important area for further research. Although low-reliability premiums for decentralized systems coincide with a low LCOE in parts of East Africa and regions just south of the Sahara Desert, in general the premium is heterogeneously distributed and does not always coincide with regions with a low LCOE, and thus it is important to consider the metrics separately.

There are a few important directions for additional investigation. First, different load profiles result in different LCOE estimates; we used a constant load profile for our study and, although we demonstrated that system costs are similar for real customers on a system in Uganda with a night-time peak demand, other realistic load profiles may lead to different costs. We report a further analysis on the impact of different load profiles in Supplementary Note 2—we found that more consumption at night increases the cost of decentralized systems. We do note, however, that our general conclusions about the reliability premium are robust to load shape. However, large commercial and industrial systems will have high costs associated with the inverter, and in this case some level of power sharing (to leverage load diversity) or direct grid connections may be essential. Second, our financial models focus on social cost and, as such, they do not include overheads for managing payment schemes, such as pay-as-you-go systems3, which bring energy access within reach in cases where the upfront costs are too large. These costs will need to be better understood and eventually factored into analyses such as ours. Third, there is a wide variation in the reported solar installation costs across SSA36; the drivers for this heterogeneity need further understanding and incorporation into planning models. Fourth, further research is needed into how the declining costs of solar and storage will impact grid tariffs; modelling these costs could push the grid parity further into the future. Finally, although preliminary analysis shows that technical failures in modern SHS are insignificant compared to resource outages in the FDS29, technical failure modes need significantly more investigation.

In the long run, if the costs we modelled can be paired with rigorous assessments of the societal benefits to electricity and reliability37, our work enables aid agencies and governments to make informed decisions about if, when and where they should rely on decentralized electrification pathways to meet reliability and development goals. However, detailed models are no substitute for the input of the end users of electricity, and it may well be that markets for decentralized solar systems will blossom long before rigorous conclusions about their benefits can be made. Centralized planning models must take into account the informal process of energy decentralization that will probably emerge in the coming years and decades.

## Methods

### Solar and load modelling

To model solar production, we used 11 years of daily average insolation incident on a horizontal surface at each location with a 1° latitudinal and longitudinal resolution. These data were obtained from the NASA Langley Research Center Atmospheric Science Data Center Surface meteorological and Solar Energy38 web portal supported by the NASA LaRC POWER Project. The data span 1 January 1995 to 31 December 2005.

To compute the mix of solar and battery storage necessary to supply power throughout the day, we modelled the system dynamics on an hourly time scale, which required up-sampling the daily average insolation to hourly average insolation. To do this, we calculated the hourly extraterrestrial horizontal insolation as a function of the day of year and location and scaled it so that the sum of hourly insolation equalled the measured daily insolation using the definitions and solar geometry equations from Duffie and Beckman39. This scaling factor, which represents cloud cover and atmospheric attenuations, is called the clearness index, and our method introduces the assumption that this is constant throughout the day, which in general is not correct39; however, given the data available, it is the simplest assumption and suffices to capture the approximate daily profile and account for seasonal variation in sunrise and sunset times.

### Computation of isoreliability curves

The isoreliability curves, or the set of solar and storage capacities that meet a specified FDS, was computed through simulation with an hourly time step. We used the isoreliability curve to represent the technical constraints imposed by the physics of the system. At a high level, the approach is to compute these constraints offline through simulation, so that they may be used in an online cost minimization.

The simulation uses the following dynamics, where Cs and Cb are the solar array and battery storage capacities, respectively, in units of kW and kWh per kWh of daily load. In and Ln are the insolation and load at hour n, in units of per unit full sun and kW per kWh daily load, respectively, which are the input vectors described above, $$\Delta P_n$$ is the excess power (or deficit if negative) that is charging or discharged from the battery at hour n, the state of charge (SOC)n is the energy stored in the battery at hour n, Un is the unmet load at hour n and N is the number of hours in the simulation (note that with an hourly time step, kW and kWh are interchangeable units):

$$\Delta P_n = C_{\rm{s}}I_n - L_n$$
(3)
$${\rm{SOC}}_{n + 1} = \max (0,\min (C_{\rm{b}},{\rm{SOC}}_n + \Delta P_n))$$
(4)
$$U_n = \max ({\rm{SOC}}_{n + 1} - {\rm{SOC}}_n - \Delta P_n,0) = \max (L_n - C_{\rm{s}}I_n - {\rm{SOC}}_n,0)$$
(5)
$${\rm{FDS}}_N = 1 - \frac{{\mathop {\sum}\limits_{n = 1}^N {U_n} }}{{\mathop {\sum}\limits_{n = 1}^N {L_n} }}$$
(6)

The simulation dynamics compute the FDS over a time horizon for a particular insolation, load shape, solar capacity and battery storage capacity. We can represent these dynamics as a map equation (7) where $$I,L \in {R}^N$$and $$C_{\rm{s}},C_{\rm{b}} \in {R}$$:

$${\rm{FDS}}_N = f_N(I,L,C_{\rm{s}},C_{\rm{b}})$$
(7)

To construct an isoreliability curve for a given location and load, we iterated across possible storage capacities and found the solar capacity that yields the target FDS. Finding this solar capacity can be written as an optimization problem, in which $$\hat{C_{\rm{b}}}$$ and $$\hat {C_{\rm{s}}}$$ are a pair of battery and solar capacities that yield the target FDS over the time horizon N, denoted $$\widetilde {{\rm{FDS}}_N}$$:

$$\hat {C_{\rm{s}}} = {\rm{arg}}\min _{C_{\rm{s}}}||\widetilde {{\rm{FDS}}_N} - f_N(I,L,C_{\rm{s}},\hat {C_{\rm{b}}})||$$
(8)

This problem is convex and can be solved with Newton’s method, to yield a point $$(\hat {C_{\rm{b}}},\hat {C_{\rm{s}}})$$on the isoreliability curve. We denote this isoreliability curve by the set of points $$C_{{\rm{FDS}},I,L}$$, noting that it is parameterized by the FDS, the insolation profile (given by the location) and the normalized load profile. We defined equation (9) as the function that gives a solar capacity on the isoreliability curve associated with a battery capacity Cb:

$$C_{\rm{s}} = G_{{\rm{FDS}},I,L}(C_{\rm{b}})$$
(9)

We used a variable step-size iteration to select storage capacity points. First, we computed the minimum storage necessary if there was an effectively infinite solar capacity, and used this as the minimum storage. Starting with this storage capacity, we computed the solar capacity, and then increased the storage capacity and repeated the sequence. We expanded the step size in storage capacity to compensate for the fact that, as storage gets large, it has a diminishing effect on reducing storage capacity. This reduces the number of iterations, but still constructs a smoothly sampled curve. Finally, we checked that the isoreliability curve was calculated with sufficient precision to be convex, and that our variable step size yielded sufficiently many points for the subsequent cost minimization to yield accurate results; if these criteria were not met, we repeated the process with greater precision in Newton’s method and a smaller step size in storage capacity. This resulted in a discretized isoreliability curve for a particular FDS at a particular location that was stored in a lookup table.

Recall that the load is normalized to 1 kWh d–1. Thus, points on the isoreliability curve are solar and storage capacities that serve a 1 kWh d–1 load at a specific FDS. We can see, by inspecting the dynamics above, that if the load was scaled by some factor, we would achieve the same reliability by scaling the solar and battery capacity by the same factor. Thus, the points on the isoreliability curve can be scaled to satisfy an arbitrary average load. The curve is dependent on the shape of the load profile, but it is independent of the average load.

We calculated and stored isoreliability curves at each degree longitude and latitude across SSA. We constructed curves for values of FDS values of 0.6, 0.8, 0.9, 0.95, 0.975, 0.9875, 0.9938, 0.9969, 0.9984, 0.9992, 0.9996, 0.9998 and 0.9999. These numbers result from the sampling expression (10) and are constructed to sample evenly from the logarithm of FDS. As we show in our results, we found that the cost of electricity scales linearly with the logarithm of FDS:

$${\rm{FDS}}_k = 1 - 0.1 \times 2^{ - k},k \in \{ - 2, - 1, \ldots ,10\}$$
(10)

This preprocessing technique allows the computationally expensive part of the algorithm—computing the isoreliability curves—to be done offline and stored for future use.

### Cost optimization

The minimum cost system design can be computed from an isoreliability curve and the costs of storage and solar. The specific costs used for different scenarios are described in the economic assumptions in the next section and are listed in Table 1. We included the discounted battery replacement costs in the total price of storage ((equation (11)), where Pb is the total price of the battery including replacement, r is the discount rate, m is the project term, T is the battery lifetime and pb is the battery costs per kWh from Table 1:

$$P_{\rm{b}} = \frac{{1 - (1 - r)^m}}{{1 - (1 - r)^T}}p_{\rm{b}}$$
(11)

The total battery capital cost was denoted Kb as in equation (12):

$$K_{\rm{b}} = C_{\rm{b}}P_{\rm{b}}$$
(12)

We also included solar derating; the simulation is performed with Cs being the derated capacity of the solar. Thus, the total capital cost of solar Ks is given by equation (13), where ps and pc are the prices per kW of solar modules plus hardware and of the charge controller, respectively, from Table 1, and α is the derating factor of 0.85 to account for dirt, wiring and conversion losses, and panel mismatch:

$$K_{\rm{s}} = C_{\rm{s}}(\frac{{p_{\rm{s}}}}{\alpha } + p_{\rm{c}})$$
(13)

The total capital cost of the system K is given by equation (14), where $$\bar L$$ is the average daily load and Kl are the capital costs per unit peak load (these are the inverter and soft costs) and Lpeak is the system peak load capacity:

$$K = (K_{\rm{s}} + K_{\rm{b}})\bar L + K_{\rm{l}}L_{{\rm{peak}}}$$
(14)

Note that Lpeak is not necessarily the peak given by the load profile used to generate the isoreliability curve; rather it is the peak capacity of the system specified by design. We calculated the LCOE using the methodology of the US National Renewable Energy Laboratory (equation (15)), where CRF is the capital recovery factor and O is the fixed annual operations and maintenance costs from Table 1:

$${\rm{LCOE}} = \frac{{K \times {\rm{CRF}} + O \times L_{{\rm{peak}}}}}{{365 \times \bar L \times {\rm{FDS}}}}$$
(15)

The CRF is given by equation (16), where r is the annual discount rate and m is the project term. We assume no variable operations and maintenance costs.

$${\rm{CRF}} = r\frac{{(1 + r)^m}}{{(1 + r)^m - 1}}$$
(16)

Our optimization problem is to minimize LCOE over the decision variables Cs and Cb, subject to the constraints of the isoreliability curve. The project term, discount rate, average load, peak load, solar derating and prices of solar and batteries are all parameters that can be adjusted independently of the isoreliability curve. This flexible parameterization without the need to perform additional simulation is what allows this approach to scale as a module of higher level optimization. Note that this problem is equivalent to minimizing the total capital costs (equation (17)) subject to the isoreliability constraint (equation (18)):

$$K_{\rm{s}} + K_{\rm{b}} = C_{\rm{s}}(\frac{{p_{\rm{s}}}}{\alpha } + p_{\rm{c}}) + C_{\rm{b}}P_{\rm{b}}$$
(17)
$$(C_{\rm{b}},C_{\rm{s}}) \in C_{{\rm{FDS}},I,L}$$
(18)

For a continuously differentiable isoreliability curve, this problem has an analytical solution given by equation (19):

$$\frac{{\partial C_{\rm{s}}}}{{\partial C_{\rm{b}}}} = - \frac{{P_{\rm{b}}}}{{\frac{{p_{\rm{s}}}}{\alpha } + p_{\rm{c}}}}$$
(19)

Given that, in the application of this algorithm, the isoreliability curve is discretized with on the order of 100 sample points, it is trivial to perform an integer search over the reliability curve to find the cost-minimizing point. We use this integer search rather than approximating the analytical solution.

### Component costs and economic assumptions

To describe the large-scale penetration of decentralized solar technologies, this study assumes that competition will drive costs to the best-practice rates. The International Renewable Energy Agency (IRENA) provides the self-reported cost breakdown of SHS greater than 1 kW installed across Africa, and includes a representative best-practice scenario that we use to construct present-day cost estimates36. This scenario indicates an installed price of US$2.3 W–1 installed, excluding the cost of storage, which is broken down into approximately US$1 W–1 for the solar modules and d.c. balance of the system costs, such as racking, wiring and circuit protection, US$0.3 W–1 for the inverter and US$1 W–1 for the a.c. balance of the system and soft costs. These estimates are consistent with IRENA’s reported values for grid-tied PV systems of a similar scale, and we used them as our best estimate of current costs.

To estimate battery costs is more complicated, as the choice of technology involves trade-offs between capital cost, efficiency, lifetime, ease of maintenance, required space and transportation logistics. In 2016, IRENA reported that most SHS use lead–acid batteries (LABs), but that lithium-ion batteries (LIBs) are beginning to appear on the market36. Based on these market trends, and on technoeconomic analyses from the literature40,41,42, we consider LIBs and LABs to be the leading battery technology choices for small-scale decentralized solar. Although the use of LIBs is still in its infancy for SHS applications larger than 1 kW, it appears that we are approximately at a parity point where a solar developer would be indifferent to the two technologies if both are available with training and support. In 2015, it was stated43 that, because of longer lifetimes, deeper depth of discharge and a better performance at high temperatures, there are cases in which LIBs are more economical. IRENA echoed this position in 201636. A simple calculation that takes these factors into account, with a 2012 LIB price of US$600 kWh–1 and a LAB price of US%120 kWh–1 (this LAB price is consistent with estimates from the literature6,36), shows that at temperatures found in Africa, LIBs result in a lower cost over their lifetime44. In 2017, for example, one could purchase a 13.5 kWh lithium ion Tesla Powerwall for USD 5,500 (which results in US$400 kWh–1) that is warranted for 10 years with a 100% depth of discharge, which indicates that LIBs are increasingly competitive with LABs.

Given the present approximate lifetime cost parity between LIBs and LABs, and the expectation that spillover effects from vehicle electrification will soon make LIBs the dominant battery technology43, we used LIB models for our analysis. This choice simplifies the comparison between present and future scenarios, and, at present, the modelled costs can be interpreted as roughly equivalent to the costs for LAB systems that are more widely available. We assumed the costs of the LIBs are currently US$400 kWh–1. We also assumed that a maximum power point tracking charge controller is used with a cost of US$0.2 W–1, which is similar to those reported36 and offered internationally by online retailers. We assumed that all the significant \$ kW–1 costs associated with the battery are captured by the charge controller and inverter costs.

To calculate the LCOE, we assumed a discount rate of 10% and a project term of 20 years. Systems may not last 20 years if they are not well maintained, but we assumed best practices. The discount rate is higher than that used by Szabó et al.20, but consistent with other studies45,46, and reflects a perception of high project risk. We assumed that battery replacement occurs at 10 years, which is conservative relative to an estimate of 15 years found in the literature47. We used the estimate to reflect uncertainty in the lifetime in potentially challenging technical environments found in decentralized systems in SSA, and to reflect the warranty term of 10 years of the Tesla battery. We also assumed replacement at the initial costs so as not to embed precise assumptions of the rate of price decreases into the model. All of these approximations are designed to be conservative, which means the actual LCOEs are likely to be lower than our estimates.

This analysis includes a look ahead at what the costs of decentralized solar could be in the future. The numbers used are meant to be suggestive, not predictive, but are drawn from market studies that are based in learning-curve methods. We constructed a future scenario based on predictions for 2025. IRENA predicts that utility-scale PV, including soft costs and balance of system will be reduced by something in the neighbourhood of 50% by 202548. We suggest that decentralized costs in Africa might fall by a similar percentage of their current costs; that is, our future scenario examines a 50% reduction in module, soft costs and balance-of-system costs (both a.c. and d.c.). We also use the ‘low-cost’ scenario published by researchers at the US National Renewable Energy Laboratory, which predicts a reduction to 25% of 2015 costs in 202547.

In calculating the trade-off between solar and storage capacity, we assumed that the inverter and soft costs are proportional to the peak power demand of the system, rather than to the solar or storage capacity. The charge controller cost is added to the cost of solar specific components, because its power rating is dependent on the output of the solar module.

The costs used in the optimization model are shown in Table 1. Soft costs include installation costs, but do not include profit or overheads for a business that provides the systems. The a.c. balance-of-system costs are assumed to be proportional to the peak load capacity of the system along with the inverter, whereas the d.c. balance-of-system costs are proportional to the array capacity.

### Tiers of access

Additional technical parameters are dictated by the ESMAP Tiers of Access30. We analysed Tier 5, which is considered the highest level of access. Tier 5 requires an average load of 8.2 kWh d–1, a peak capacity of 2 kW and a service available for 159 out of 168 hours in a week, which translates to an FDS of 94.6%30. These parameters were used to calculate the LCOE using the optimization methods described above.

### Empirical grid and electrification metrics

We used the World Bank Enterprise Surveys on Infrastructure to estimate the ASAI for countries in SSA28. These surveys estimate the number of grid outages per month and the average duration of the outage. From this, we can compute the average number of hours in a year when the grid power is out for each country, and then divide by the number of hours in a year to give the ASAI. At the time of writing, the years surveyed ranged from 2006 to 2017 by country.

Electricity tariff data were obtained at a national level from the World Bank and are available for most countries34. We used retail rates. Where tiered rate structures are in effect, we used the rate at 250 kWh per month for ESMAP Tier 5.

National electrification rates are available through the World Bank data tables35. Supplementary Table 1 gives the tariff, reliability and electrification metrics.

### Code availability

Analysis was conducted using MATLAB R2017a. The code is available upon request from J.T.L. at jtlee@berkeley.edu and at https://github.com/leejt489/solar-reliability-cost-matlab. A Python version is also available upon request and at https://github.com/leejt489/solar-reliability-cost-python. An implementation of the model is available at https://emac.berkeley.edu/reliability.

## Data availability

All of the data used in this study, with the exception of the sample microgrid load profile from Uganda, is publicly available and referenced here. The sample microgrid load profile is owned by New Sun Road, P.B.C., and can be made available upon reasonable request. Tables on national electrification rates35, grid reliability28 and electricity tariffs34 used to generate Figs. 1 and 6 are included in Supplementary Table 1 for convenience. Solar insolation data are available from the US National Aeronautics and Space Administration38, and the code to download and analyse this data is available at the repositories listed above.

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## Acknowledgements

We thank the US National Science Foundation for supporting this work through the CyberSEES program (award no. 1539585) and are grateful to J. P. Carvallo, R. Shirley, D. Kammen and I. Ferrall for their advice and comments. We also thank J. Sager and New Sun Road, P.B.C., for sharing their insights into decentralized system designs and sample data from their systems.

## Author information

J.T.L. contributed the primary development of the optimization program, methods and computer modelling. D.S.C. supervised the research effort. The authors jointly developed the research questions, analysis, conclusions and manuscript.

Correspondence to Duncan S. Callaway.

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### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary Notes 1–4, Supplementary Figures 1–5, Supplementary Tables 1–4, Supplementary References

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### Cite this article

Lee, J.T., Callaway, D.S. The cost of reliability in decentralized solar power systems in sub-Saharan Africa. Nat Energy 3, 960–968 (2018) doi:10.1038/s41560-018-0240-y

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