Design of stand-alone electrification systems using fuzzy mathematical programming approaches

1 Design of stand-alone electrification systems using fuzzy mathematical programming approaches R. Galleguillos-Pozo1, B. Domenech1,2,*, L. Ferrer-Martí1,3, R. Pastor1,2 1 Supply Chain and Operations Management Research Group, Universitat Politècnica de Catalunya – BarcelonaTech 2 Department of Management, Universitat Politècnica de Catalunya – BarcelonaTech 3 Department of Mechanical Engineering, Universitat Politècnica de Catalunya – BarcelonaTech Corresponding author (*): [email protected] ; Av. Diagonal 647, Room H-11.46, Barcelona (Spain) Abstract Currently, around 1.1 billion people lack access to electricity, mainly in developing countries. Solar photovoltaic systems can provide electricity, but the design is complex, having to size and site the equipment. Besides, since forecasting consumption habits for newly electrified populations is complex, the estimation of the electricity needs is uncertain. This work addresses, for the first time, demand uncertainty to assist promoters in designing electricity access projects, simultaneously solving the sizing and siting problems, using fuzzy logic. In particular, a mathematical model is proposed, introducing uncertainty through five modelling approaches with different fuzzy logic assumptions: three based on the literature and two novel ones according to the conflicting problem nature (project cost minimisation vs electricity supply maximisation). The five approaches are compared and the new ones obtain solutions achieving a higher satisfaction regarding the cost and electricity supply. Then, the most efficient approach is applied in two Peruvian communities, comparing the solutions with those obtained without uncertainty. The results of the proposal show a better balance between the project cost and the demand supplied. Hence, the proposal can help promoters in developing countries to better design electricity access projects where the demand estimation is complex. Keywords: isolated distribution networks; rural electrification; fuzzy; MILP model 2 1. Introduction Currently, around 1.1 billion people worldwide still lack electricity access, mainly in rural areas of developing countries [REN21, 2017]. Stand-alone systems based on renewable energy are suitable for supplying electricity in these regions, since they can save costs regarding grid extension while avoiding external dependencies [Chauhan & Saini, 2014]. In addition, some authors highlight the importance of renewable energy technologies for achieving the Sustainable Development Goals [Kuriqi et al., 2020]. Among the existing technologies, hydropower plants have high technoeconomic performance despite pluvial variations [Kuriqi et al., 2019], although they are subject to having an existing river waterfall nearby. In contrast, solar photovoltaic (PV) takes advantage of solar radiation to produce electricity, even under cloudy conditions, and is technically simple, cheap and well-known, making it suitable in rural areas of developing countries [Ellabban et al., 2014]. For instance, the economic feasibility of this technology has been demonstrated in SubSaharan Africa [Okoye & Oranekwu-Okoye, 2018]. Besides, advances in the prediction of solar radiation make the projects’ design robust in the event of resource variations [Karasu & Altan, 2019]. Individual PV supplies have played a key role in providing electricity access [Ridha et al., 2020], given the usual dispersion between consumption points such as houses, schools or health centres. However, the combination of individual supplies and isolated distribution networks can achieve significant benefits [Domenech et al., 2014]. Isolated distribution networks, i.e., a single generation point from which several consumption points are supplied, can save costs through economies of scale while improving supply quality, favouring equal consumption among users and enabling flexibility in the event of occasional demand increases [Moretti et al., 2019]. When addressing the design of rural electrification projects, both the sizing and siting problems must be addressed [Gamarra & Guerrero, 2015]. The sizing aims to determine the size of the generators and the storage system to supply electricity, according to the demand, the resources available and technical aspects such as equipment efficiencies. The siting problem focuses on locating equipment, depending on the concentration/dispersion of demand and the variability of resources, as well as the design of distribution lines from the generators to end-users. In order to support decisionmaking, several optimisation tools have been developed [Domenech et al., 2019]. Concerning the sizing tools, several options have been developed in the literature for energy planners [Ferrari et al., 2019]. For instance, HOMER is a well-known software application that includes many options for electricity generation and storage [Sinha & Chandel, 2014]. This software was used in India to study several renewable energy options for different rural applications, combining technologies to improve supply security and achieve cheap and technically efficient solutions [Sen & Bhattacharyya, 2014]. Concerning the siting tools, ViPOR addresses the problem of defining the best location for generation points, the structure of isolated distribution networks and the individual users [Lambert & Hittle, 2000]. Nevertheless, ViPOR has technical limitations in defining networks, while integrating HOMER and ViPOR to achieve a comprehensive tool is not straightforward [Domenech et al., 2019]. Alternatively, Ferrer-Martí et al. [2013] develop a Mixed Integer Linear Programming (MILP) model that simultaneously addresses the sizing and siting problems. This model is used to design electrification projects in Peru, defining the minimum cost size and location 3 of equipment, including the network structure and individual supplies. Ranaboldo et al. [2014] develop a heuristic procedure to accelerate the solution of the above MILP model, achieving results in a shorter calculation time. Domenech et al. [2015] extend the research from Ferrer-Martí et al. [2013], including the above MILP model in a multicriteria decision-making process for the comprehensive design of electrification projects. Finally, García-Villoria et al. [2020] develop a heuristic procedure that allows management constraints to be included in the design of electrification projects. In general terms, the reviewed tools aim to determine the cheapest combination of technologies to satisfy the electricity requirements of the population [Sinha & Chandel, 2014]. The energy and power demand is generally assumed as a unique (deterministic) value and the results, particularly the project cost, are subject to the quality of their estimation. However, determining the demand is not straightforward and is subject to uncertainty. The previously non-electrified population is accustomed to a supply based on kerosene or batteries, mainly for lighting and small telecommunication devices (radio, mobile phone, etc.). In contrast, electricity can significantly modify habits, extending productive/study hours for adults/children, accessing new devices (TV, fridge, etc.) or developing formerly inconceivable products. Surveys and interviews are carried out to determine the demand of the target community, along with meetings with population groups (women, men, children, elders, etc.) to get an overall perspective of their expectations. Despite this, quantifying the demand is complicated and the equipment available usually has staggered characteristics, so small variations in the demand can have a significant impact on the solution obtained, and particularly the cost. Fuzzy-based models have achieved realistic solutions to complex problems, particularly in renewable energy systems under uncertainty [Suganthi et al., 2015]. The use of fuzzy logic to improve the response of controllers in different industry applications has been researched from an electrical perspective [Aslan et al., 2017]. Regarding the design of energy systems using multicriteria tools, Onar et al. [2015] develop a multi-expert decision framework to assist investors in choosing the appropriate wind energy technology, dealing with the decision-makers’ uncertainty when defining some input parameters. Dincer & Yuksel [2019] compare different fuzzy approaches for modelling uncertainty in decisionmaking for investments in renewable energy systems. Regarding optimisation tools, Li et al. [2010] introduce fuzzy and stochastic elements into a MILP model for planning regional-scale energy and environmental systems in order to manage uncertainties in the energy demand. Ammar et al. [2010] develop a neuro-fuzzy algorithm for the management of a household PV panel, balancing the predicted generation and end-user electricity requirements in order to improve the efficiency of the system. Hu et al. [2011] propose a fuzzy model for planning a regional electric power generation system and provide insights regarding the influence of uncertain parameters on the final solution. Lamedica et al. [2018] include demand uncertainty through stochastic simulation in the design of a wind-PV energy system. Yu et al. [2019] develop an interval possibilistic-stochastic model, which allows the simultaneous consideration of uncertainty in different input parameters, for planning the energy mix supply of Qingdao (China). Mohammadi et al. [2020] use fuzzy sets to take uncertainty in the estimation of energy demand and wind power generation into account, to improve the robustness of the proposed solution. Finally, Wang et al. [2020] examine the inclusion of demand uncertainty in the design of largescale distributed energy systems. They compare an a priori approach, combining two stochastic models, with an a posteriori approach, combining a Monte Carlo simulation and a deterministic model. 4 1.1. Motivations, objectives and contributions of the study As observed in the above literature review, several tools for the design of rural electrification projects have been developed, focusing on the sizing or the siting problems, while the simultaneous solution of both is less researched. Regarding the literature including fuzzy logic in the design of energy systems (see previous paragraph), some address uncertainty in decision-makers’ preferences [Aslan et al., 2017] or in the whole decision-making process [Onar et al., 2015]. Others address demand uncertainty, but focussing on individual facilities [Ammar et al., 2010], industrial services [Lamedica et al., 2018] or regional-scale projects examining demand as a whole, such as Li et al. [2010], Hu et al. [2011], Yu et al. [2019], Mohammadi et al. [2020] and Wang et al. [2020]. In contrast, to the best of the authors knowledge, the design of electricity access projects, combining the sizing and siting problems, while assessing uncertainty in the estimation of the electricity demand for newly electrified populations has not be studied. In this regard, Karunathilake et al. [2019] state that additional research is required on the planning of community-level projects, taking the typical uncertainty in such systems into account. In particular, studies must go beyond economic issues and adapt their solutions to local realities [Weinand et al., 2020]. In this context, the objective of this work is to improve the design of PV-based rural electrification by dealing with the uncertainty in the estimation of demand. For this purpose, a fuzzy mathematical model is proposed to assist promoters in developing countries, which simultaneously addresses the sizing and siting of equipment. In particular: 1. As starting point for the fuzzy model, first, a deterministic model to design rural electrification projects is developed, taking into account very specific details of the local communities, such as the location of end-users and their energy and power requirements, except uncertainty. The model allows sizing the generation equipment (PV panels and inverters) and the electricity distribution configuration (individual systems or isolated distribution networks, depending on the dispersion of consumption points) to be determined simultaneously. This deterministic model is the base to develop the fuzzy models and, moreover, will be used to generate the input data for the fuzzy models. 2. As the best modelling approach to include fuzziness is not straightforward, five fuzzy MILP (FMILP) models are developed and compared. These models use different approaches to include fuzziness through satisfaction variables for the energy and power demands and, consequently, the cost. As a result, each approach balances, in a different manner, the end-users’ satisfaction regarding the project cost, energy and power supply. The first three approaches are based on the literature review and the other two are completely new. Moreover, the approaches are replicated, studying two assumptions that have not been considered in the literature: minimum satisfaction (the focus is on the least satisfied consumption point) and average satisfaction (the focus is on all the consumption points globally). 3. In order to identify the best fuzzy model, their performance is evaluated through a computation experiment. The experiment compares the results of the 5 approaches, considering the 2 assumptions as well as the 5 ranges of uncertainty regarding the demand estimation of end-users. The results show that approach 4, which directly balances the cost vs the global demand, obtains very good values in most instances. In addition, it is a simple option which facilitates the analysis of results for project promoters. 5 4. Finally, the fuzzy models are validated in two real communities located in the Peruvian highlands. Consequently, the solutions obtained with approach 4 can be compared with the solutions that would have been obtained without including fuzziness (hence estimating the demand in a deterministic manner). Results show that fuzzy solutions achieve an overall good satisfaction for cost and demand in comparison with non-fuzzy solutions. Hence, the proposed fuzzy model provides rural electrification promoters in developing countries with an easy way to design electricity access projects, where the demand is particularly complex to assess. The paper is organised as follows. In Section 2, the problem is described technically and in terms of uncertainty in demand. In Section 3, the deterministic MILP model to design PV-based electrification systems is developed and then five fuzzy modelling approaches are proposed. In Section 4, a computation experiment is carried out to compare the FMILP approaches, and the best one is applied to real case studies in Section 5. Finally, in Section 6, the main conclusions are summarised. 2. Problem description of photovoltaic-based electrification systems This section presents the technical considerations of PV-based electrification systems (Section 2.1), the way that electricity demand is estimated (Section 2.2) and the membership functions to include uncertainty in demand and, consequently, the project cost (Section 2.3). 2.1. Technical considerations Figure 1 shows the scheme of elements included in PV-based electrification systems, whose main characteristics and relationships are as follows:  Consumption points: they can be houses, schools, health centres or any other point needing electricity. Each one has an energy and power requirement.  PV generation: PV panels transform solar radiation into electricity. Important studies have been carried out to estimate solar radiation depending on several climatic parameters [Karasu et al., 2017]. However, the solar resource is assumed to be uniform within a community [Gueymard & Wilcox, 2011]. PV panels can only be located at consumption points and there is a limit on the number to be installed at any one point.  Inverters: these devices transform the direct current (DC) leaving the panels into alternating current (AC), which is more suitable for most electrical appliances.  Electricity distribution: electricity is distributed through low voltage (LV) lines to consumption points individually or as isolated distribution networks, which have a radial scheme and are more suitable (i.e., technically simpler and cheaper) for rural areas of developing countries [Lambert & Hittle, 2000]. 6 Figure 1 – PV electrification system with isolated distribution network distribution [adapted from Ferrer-Martí et al., 2013] 2.2. Electricity demand estimation Assessing a community to be electrified is a complex task and includes quantitative and qualitative information about its characteristics and population, the energy sources before electrification and the possible future uses of electricity. The ultimate goal is to define the energy and power demand of each end-user. This laborious process is commonly carried out without standardised methodologies, as it depends on each case study [Gupta, 2003]. Next, some important issues are summarised. In order to get information about a community, some tools can help the interaction with inhabitants while identifying the needs of each person and avoiding pressure from leaders or outside influences. Information from local and regional databases must be considered, but also data gathered from surveys, individual interviews and meetings with population groups (women, men, children, etc.). Regarding the surroundings, the climatology [Schäfer et al., 2011], which influences the population’s habits must be taken into account, as well as nearby communities [Camblong et al., 2009] which provide indications about future energy use. Issues to be examined inside the target community include: current productive activities and future expectations [Anderson & Doig, 2000]; the configuration of houses and other consumption points which determine demand [Camblong et al., 2009], for instance, the number of rooms. Finally, the energy sources prior to electrification (candles, batteries, etc.) and their usage (daily hours, season, etc.) must be analysed [Schäfer et al., 2011]. With this information, the daily energy demand (linked to the amount of electricity that can be consumed throughout the day) and the peak power (linked to the devices that can be simultaneously connected) must be summarised in a unique value. As mentioned before, this is not straightforward and is subject to uncertainty, even more so when assuming the value should be valid for the project’s entire lifetime. In addition, the staggered size of equipment and economies of scale can cause a slight increase in demand, which has a significant project cost growth, or a small budget increase can allow more powerful equipment to be used. For instance, if two PV panels are considered (energy: 50 and 100 Wh/day), 7 a deterministic energy demand of 51 Wh/day would require the second and more expensive panel. In contrast, a 50 Wh/day value could be supplied using the first option and the cost would significantly diminish. Consequently, even if the energy and power demands are the origin of uncertainty, the project cost is also uncertain. From the above analysis of the population and their expected consumption patterns, an essential and improved demand can be determined. The essential is an energy and power demand below which the project would not be advisable because end-user expectations would not be satisfied. In contrast, the improved is a demand above which the excess would not be used, as end-users would have more electricity than needed. Between both values, a balance should be sought between expensive solutions with a high supply and cheap solutions with a low supply. To do so, in the next section, three membership functions are defined to represent end-users’ satisfaction with regard to cost, energy and power. 2.3. Membership functions A common strategy for dealing with fuzzy environments, such as happens with the cost, energy and power, is through membership functions [Bilgen, 2010]. In this paper, such functions are calibrated by the direct method [Mendel & Korjani, 2018], and through consultations to rural electrification promoters and local population, hence defining: the full non-membership as the essential demand (since any supply below this threshold does not cover the basic needs of population); the full membership as the improved demand (since any supply above this threshold is not used because population expectations have been covered yet); and the crossover point as a linear progression (since it represents the evolution of end-users’ satisfaction when the supply increases from the essential to the improved demand). By analogy, the same behaviour is considered for the cost function and the functions are normalised in a 0-1 scale to be able to compare the cost vs the energy and power. 2.3.1. Cost membership function A deterministic MILP can be formalised to obtain the minimum cost of the project (CMIN) when considering the essential demand as input data and, when the improved demand is considered, the maximum cost of the project (CMAX). Hence, for solutions whose cost is lower than or equal to the minimum value, the maximum satisfaction is attained (λC = 1); while for solutions equal to or above the maximum cost, the minimum satisfaction is achieved (λC = 0). For intermediate values, a linear progression from 1 to 0 is assumed, where x represents the cost of the studied solution. Equation A and Figure 2 represent the cost membership function ( () Cx   ) mathematically and graphically, respectively. 1 () 0 MIN MAX MIN MAX CMAX MIN MAX if x C Cx x if C x C CCif C x             (eq. A) 8 Figure 2 – Cost membership function 2.3.2. Energy membership function As explained before, in a deterministic scenario the energy demand is estimated as a unique value. However, this estimation might be subject to some uncertainty, so the energy requirements of the population may lie within a range from an essential (EMIN) to an improved (EMAX) value. Hence, for solutions whose energy is lower than or equal to the minimum value, the minimum satisfaction is attained (λE = 0); while for solutions equal to or above maximum energy, the maximum satisfaction is achieved (λE = 1). For intermediate values, a linear progression from 0 to 1 is assumed, where x represents the energy of the studied solution. Equation B and Figure 3 represent the energy membership function ( () Ex   ) mathematically and graphically, respectively. 0 () 1 MIN MIN MIN MAX EMAX MIN MAX if x E xE x if E x E EEif E x             (eq. B) Figure 3 – Energy membership function 9 2.3.3. Power membership function The power membership is analogous to the energy membership function, but logically considering its own range of values (PMIN to PMAX) between the essential and improved power demand requirements of the population. Equation C and Figure 4 represent the power membership function ( () Px   ) mathematically and graphically, respectively. 0 () 1 MIN MIN MIN MAX PMAX MIN MAX if x P xP x if P x P PPif P x             (eq. C) Figure 4 – Power membership function As might be expected, there is an opposite tendency regarding satisfaction: the lower the cost satisfaction, the higher that of energy and power; the higher the cost satisfaction, the lower the energy and power satisfaction. Therefore, it is interesting to examine a balance of cost, energy and power in order to achieve the highest possible global satisfaction. Different approaches can be conceived when defining the global satisfaction, as proposed in the next section. 3. Mathematical models In this section, first, the deterministic model to design rural electrification projects is developed, taking into account very specific details of the local communities, such as the location of end-users and their energy and power requirements, except uncertainty (Section 3.1). This model considers a deterministic value for the energy and power demands and, therefore, the project cost. Next, five fuzzy MILP (FMILP) approaches are proposed to include uncertainty in the estimation of demand, thus balancing end-users’ satisfaction with regard to cost, energy and power (Section 3.2). In particular, three approaches are developed based on fuzzy logic methodologies from the literature (approach 1 according to Zimmermann [1976] and approaches 2 and 3 according to Werners [1987]), while two approaches (4 and 5) are developed according to the specific characteristics of the target problem. 16 In contrast, in the average satisfaction assumption, the mean is calculated from the project cost satisfaction (λC) and the average energy (𝜆𝐸     =∑𝜆𝐸𝑝 𝑁 𝑝=1 𝑁) and power (𝜆𝑃     =∑𝜆𝑃𝑝 𝑁 𝑝=1 𝑁) of all consumption points p. In addition, the mean is weighted using parameter γ, which represents the compensation between the lowest global satisfaction and the mean satisfaction [Bilgen, 2010]. Hence, for γ = 1 all the importance is assigned to the lowest global satisfaction, resulting in approach 1. In contrast, when γ diminishes, progressive importance is assigned to the mean satisfaction, up to assigning the same importance to both criteria for γ = 0. However, the mean satisfaction is always conceived as a secondary criterion, so approach 3 is proposed.  Approach 3 aims to balance the lowest global satisfaction and the mean satisfaction ((1.3i) and (1.3ii)). Hence, the parameter γ can be adjusted according to the decision-maker’s preferences, depending on whether more importance is assigned to the lowest global satisfaction (high γ values) or to the mean satisfaction (low γ values). For γ = 1.0 the objective function is as in approach 1; for γ = 0.5 it is equivalent to approach 2 with γ = 0.0; and for γ = 0.0 the function balances the cost, energy and power satisfaction with the same importance. However, the same relevance is considered for cost, energy and power which, as explained next, might be confusing; so approach 4 is proposed.  Approach 4 changes the balancing concept. Approaches 2 and 3 balance the lowest global satisfaction and the mean satisfaction. However, while the cost is of “the lower, the better” type, the energy and power demands are of “the higher, the better” type. Consequently, balancing the three aspects at the same level might mislead results. Therefore, this approach balances the cost satisfaction (prioritising cheap solutions) versus the energy and power (prioritising high supply solutions). However, no compensation parameter is considered, so approach 5 is proposed.  Approach 5 starts from the same concept as approach 4, but includes a compensation parameter γ. For γ = 1.0 all importance is assigned to cost; while for γ = 0.0 all importance is assigned to energy and power. Note that for γ = 0.5 the objective function is equivalent to approach 4. Concerning the compensation parameter γ from approaches 2, 3 and 5, a value of 0.6 is considered, as in the literature [Mula et al., 2006]. 17 Table 2 – Objective functions of the five fuzzy modelling approaches Approach Minimum satisfaction assumption Average satisfaction assumption 1   MAX Z  (1.1i)   MAX Z  (1.1ii) 2     1 (1 ) 3 MAX Z C E P             (1.2i)   11 1 (1 ) 3 NN pp pp EP MAX Z C N                    (1.2ii) 3     1 (1 ) 3 MAX Z C E P               (1.3i)   11 1 (1 ) 3 NN pp pp EP MAX Z C N                      (1.3ii) 4     1 2 MAX Z C E P        (1.4i)   11 1 2 NN pp pp EP MAX Z C N         (1.4ii) 5       1 12 MAX Z C E P             (1.5i)     11 1 12 NN pp pp EP MAX Z C N              (1.5ii) 18 4. Computation experiment In this section, a computation experiment is performed in order to compare the performance of the 3 FMILP approaches developed based on fuzzy logic methodologies from the literature and the 2 FMILP approaches developed according to the specific characteristics of the target problem. 4.1. Data for the computation experiment The computation experiment is based on data from a real case study [Ferrer-Martí et al. 2013]. An instance is randomly generated, reducing the number of consumption points to facilitate analysis of the results and better understand the influence of the FMILP approaches on the solutions obtained. The input parameters are:  Consumption points: 4 houses (essential energy / power demand, EpMIN / PpMIN: 280 Wh/day / 200 W), 1 health centre (essential energy / power demand, EpMIN / PpMIN: 975 Wh/day / 600 W) and 1 school (essential energy / power demand, EpMIN / PpMIN: 975 Wh/day / 1000 W).  PV panels: 4 options (cost, CSs: $450, $635, $820 and $1000; energy, ESs: 220, 325, 435 and 650 Wh/day; maximum number that can be installed at a point, NS: 40).  Inverters: 4 options (cost, CIi: $375, $1200, $1800 and $2300; power, PIi: 300, 1200, 2000 and 3000 W).  Electricity distribution lines: 1 option (cost, CC: $5/m; maximum distance to connect 2 points, Lmax: 1000 m). Concerning the demand increases (ΔEp and ΔPp) five scenarios are considered: increases of 10%, 20%, 30%, 40% and 50% for both the energy and power. From these values, the maximum (CMAX) and minimum (CMIN) project costs are calculated using the deterministic MILP model. Hence, a total of 50 instances are solved (5 FMILP approaches, 2 assumptions and 5 scenarios). The experiment is carried out with software ILOG CPLEX Optimization Studio 12.6, on a computer of 2.00 GHz, Intel Core i3-6006U CPU, with 6.00 GB of RAM. 4.2. Results of the computation experiment Optimal solutions are achieved for all the instances in a very short calculation time (a few seconds for each case). Table 3 shows the lowest satisfaction variable (λ) and the satisfaction variables for cost (λC), energy (𝜆𝐸 for the minimum satisfaction assumption and 𝜆𝐸     for the average satisfaction assumption) and power (𝜆𝑃 and 𝜆𝑃     ). A first examination of solutions shows that, in global terms, approaches 2 and 3 achieve the same or a slightly smaller lowest satisfaction variable (λ) than approach 1 (0.386 vs. 0.418 in the worst case); however, they ensure significantly higher values (that can exceed more than 0.5 difference) for the other satisfaction variables. Consequently, although approach 1 is the simplest in terms of modelling, it does not allow the three studied aspects of the problem (cost, energy and power) to be evaluated. 19 Table 3 – Results of the computation experiment for the 50 instances solved Scenario Minimum satisfaction assumption Average satisfaction assumption Approach Approach 1 2 3 4 5 1 2 3 4 5 10% λ 0.484 0.484 0.484 - - 0.484 0.484 0.484 - - λC 0.484 0.564 0.564 0.401 1.000 0.484 0.564 0.564 0.894 1.000 𝝀𝑬 | 𝝀𝑬     0.484 0.484 0.484 0.878 0.484 0.484 0.570 0.570 0.754 0.754 𝝀𝑷 | 𝝀𝑷     0.484 1.000 1.000 1.000 0.001 0.484 1.000 1.000 0.833 0.612 20% λ 0.577 0.577 0.577 - - 0.536 0.577 0.577 - - λC 0.577 0.577 0.577 0.577 0.848 0.536 0.577 0.577 0.941 0.941 𝝀𝑬 | 𝝀𝑬     0.577 0.804 0.804 0.804 0.056 0.536 0.884 0.884 0.538 0.538 𝝀𝑷 | 𝝀𝑷     0.577 1.000 1.000 1.000 1.000 0.536 1.000 1.000 0.834 0.834 30% λ 0.550 0.550 0.536 - - 0.550 0.536 0.536 - - λC 0.550 0.577 0.678 0.678 0.884 0.550 0.678 0.678 0.886 0.886 𝝀𝑬 | 𝝀𝑬     0.550 0.550 0.536 0.536 0.037 0.550 0.559 0.559 0.577 0.577 𝝀𝑷 | 𝝀𝑷     0.550 1.000 1.000 1.000 1.000 0.550 1.000 1.000 0.833 0.833 40% λ 0.425 0.413 0.402 - - 0.425 0.402 0.402 - - λC 0.425 0.677 0.754 0.754 0.911 0.425 0.426 0.426 0.861 0.861 𝝀𝑬 | 𝝀𝑬     0.425 0.413 0.402 0.402 0.028 0.425 0.788 0.788 0.620 0.620 𝝀𝑷 | 𝝀𝑷     0.425 0.909 0.909 0.909 0.909 0.425 1.000 1.000 0.800 0.800 50% λ 0.418 0.418 0.418 - - 0.418 0.418 0.386 - - λC 0.418 0.431 0.431 0.691 0.923 0.418 0.431 0.386 0.785 0.785 𝝀𝑬 | 𝝀𝑬     0.418 0.418 0.418 0.321 0.021 0.418 0.612 0.824 0.567 0.567 𝝀𝑷 | 𝝀𝑷     0.418 1.000 1.000 1.000 0.727 0.418 1.000 1.000 0.900 0.900 4.3. Discussion about the computation experiment In order to discuss the results obtained with the 5 FMILP approaches, the satisfaction variables (λC, λE, 𝜆𝐸     , λP and 𝜆𝑃     ) are compared with each other. To do so, for each demand scenario, the values achieved for each approach are divided by the maximum value that can be obtained for any approach and then multiplied by 100. Table 4 shows the comparison. For instance, in the average satisfaction assumption and 20% scenario, the maximum cost satisfaction for the five approaches is 0.941, so approaches 4 and 5 obtain a value of 100%, while approaches 2 and 3 get 61.3% (0.577/0.941*100) and approach 1 gets 57.0% (0.536/0.941*100). Moreover, for the sake of clarity, the results are coloured on a triple-scale:  Green: very good value, equal to or less than 20% difference with regard to the best value.  Orange: good value, equal to or less than 40% difference with regard to the best value.  Red: bad value, more than 40% difference with regard to the best value. 20 The last row in Table 4 shows the total sum of percentages. The higher this value the better the approach, as it globally achieves better (or closer to better) satisfaction for cost, energy and power for all scenarios. Table 4 – Comparison between the instances solved for each approach (%) Scenario Minimum satisfaction assumption Average satisfaction assumption Approach Approach 1 2 3 4 5 1 2 3 4 5 10% λC 48.4% 56.4% 56.4% 40.1% 100% 48.4% 56.4% 56.4% 89.4% 100% 𝝀𝑬 | 𝝀𝑬     55.1% 55.1% 55.1% 100% 55.1% 64.2% 75.6% 75.6% 100% 100% 𝝀𝑷 | 𝝀𝑷     48.4% 100% 100% 100% 0.1% 48.4% 100% 100% 83.3% 61.2% 20% λC 68.0% 68.0% 68.0% 68.0% 100% 57.0% 61.3% 61.3% 100% 100% 𝝀𝑬 | 𝝀𝑬     71.8% 100% 100% 100% 7.0% 60.6% 100% 100% 60.9% 60.9% 𝝀𝑷 | 𝝀𝑷     57.7% 100% 100% 100% 100% 53.6% 100% 100% 83.4% 83.4% 30% λC 62.2% 65.3% 76.7% 76.7% 100% 62.1% 76.5% 76.5% 100% 100% 𝝀𝑬 | 𝝀𝑬     100% 100% 97.5% 97.5% 6.7% 95.3% 96.9% 96.9% 100% 100% 𝝀𝑷 | 𝝀𝑷     55.0% 100% 100% 100% 100% 55.0% 100% 100% 83.3% 83.3% 40% λC 46.7% 74.3% 82.8% 82.8% 100% 49.4% 49.5% 49.5% 100% 100% 𝝀𝑬 | 𝝀𝑬     100% 97.2% 94.6% 94.6% 6.6% 53.9% 100% 100% 78.7% 78.7% 𝝀𝑷 | 𝝀𝑷     46.8% 100% 100% 100% 100% 42.5% 100% 100% 80.0% 80.0% 50% λC 45.3% 46.7% 46.7% 74.9% 100% 53.2% 54.9% 49.2% 100% 100% 𝝀𝑬 | 𝝀𝑬     100% 100% 100% 76.8% 5.0% 50.7% 74.3% 100% 68.8% 68.8% 𝝀𝑷 | 𝝀𝑷     41.8% 100% 100% 100% 72.7% 41.8% 100% 100% 90.0% 90.0% Total percentage 947.2% 1263.0% 1277.8% 1311.4% 953.2% 836.1% 1245.4% 1265.4% 1317.8% 1306.3% Finally, to ease the comparison of approaches, the results from Table 4 are summarised in Figure 6. For each FMILP approach the amount of very good (green), good (orange) and bad (red) results is shown, distinguishing between the minimum (min) and the average (ave) assumptions. The following conclusions can be gathered:  Approach 1. Obtains few very good values (4) and the maximum bad values (20).  Approach 2. Obtains very good values (17) but also bad ones (6).  Approach 3. Obtains very good values (19) but also bad ones (6).  Approach 4. Obtains very good values in most instances (22) and only 1 bad value.  Approach 5. Obtains very good values (19) but also bad ones (6). In particular, it obtains some very bad values for the minimum satisfaction assumption (less than 10% in 4 cases). 21 Figure 6 – Summary of the comparison of the instances solved for each approach As observed from the results discussion, the approaches based on the literature (1, 2 and 3) obtain the worst results. Approach 1 is definitely not suitable, since it only focusses on the worst satisfaction among cost, energy and power, disregarding the other two. Approaches 2 and 3 consider both the worst satisfaction and the average of the three satisfactions. This might be suitable for other problems [Bilgen, 2010], but does not respond to the nature of the problem studied in this paper: the worst satisfaction is considered twice (as the worst satisfaction and in the average satisfaction), distorting results. In contrast, approaches 4 and 5, specifically developed regarding the nature of this problem, balance cost minimisation better (tending towards cheap and low-demand solutions) vs energy and power maximisation (tending towards expensive and high-demand solutions). In particular, approach 4 (which corresponds to approach 5 with γ=0.5) obtains the highest performance: the maximum of very good values, the minimum of very bad ones and the highest values for the total percentage; i.e., it is globally better or closer to better satisfactions for cost, energy and power. Consequently, this is the approach considered for the case study application. 5. Case study application Section 2.2 describes the complexity of the demand estimation for populations with recent access to electricity. In spite of that, the only manner existing in the literature to design rural electrification projects considering uncertainty in demand is by executing the models and tools several times, each one with a different demand value. This process can be long and confusing when defining the demand values to be tested. In contrast, the proposed fuzzy models aim to overcome this by directly finding a balance between a range of demand values. Hence, project promoters are only asked about an essential demand (below which end-user expectations would not be satisfied) and an improved demand (above which the excess would not be used by end-users); the FMILP directly returns the most balanced option. Therefore, in this section, two case studies are solved to compare the solutions of the deterministic MILP models (which would have been obtained prior to 22 this paper) with the fuzzy FMILP models, which directly find a balanced solution. The application focusses on approach 4, which obtained the best results in Section 4 and is a simple option, which directly compares the cost, energy and power demands without weighting or calibration parameters, facilitating analysis of the results for project promoters. 5.1. Data for the case study application The case study application focusses on the real communities of El Alumbre and Alto Peru (Peru), promoted by the NGOs Practical Action (Peru), Engineering Without Borders (Spain) and Green Empowerment (USA) [Ferrer-Martí et al., 2013]. Figure 7 shows their location in Cajamarca (Peru). Figure 7 – Location of El Alumbre and Alto Peru in the region of Cajamarca (Peru) [Ferrer-Martí et al., 2011] As input data, first the location of all consumption points is considered in both communities: El Alumbre, 33 houses, a school and a health centre; Alto Peru, 26 houses. Moreover, the equipment from Section 4.1 is completed with additional intermediate-size options to ease the generation of possible solutions. The instances are solved with software ILOG CPLEX Optimization Studio 12.6, on a computer of 2.00 GHz, Intel Core i3-6006U CPU, with 6.00 GB of RAM. Optimal solutions are always achieved in less than 3600 s. 23 5.2. Results of the case study application Table 5 shows the results of the case study application. The two communities are dealt with by the deterministic MILP model considering the essential, EpMIN and PpMIN from Section 4.1, and improved, 50% higher, demands (columns 1 and 4, respectively). Next, the two communities are solved with the FMILP approach 4, for the minimum and average satisfaction assumptions (columns 2 and 3, respectively). The FMILP is solved considering the essential and improved demands used in the MILP models, so the MILP solutions provide the minimum cost (CMIN) and the cost range (ΔC) required for the FMILP as input data (Section 3.2.1). By rows, Table 5 details the cost, energy and power satisfaction for the solutions in both communities studied: Table 5 – Comparison between the instances solved for each approach Case study MILP essential demand solution FMILP minimum satisfaction assumption FMILP average satisfaction assumption MILP improved demand solution El Alumbre Cost Cost $35614 $39799 $38034 $46174 λC 1.00 0.60 0.77 0.00 Energy min{𝝀𝑬𝒑} 0.00 0.32 0.00 1.00 max{𝝀𝑬𝒑}–min{𝝀𝑬𝒑} 0.64 0.13 1.00 0.00 𝝀𝑬     0.30 0.33 0.30 1.00 Power min{𝝀𝑷𝒑} 0.00 1.00 0.00 1.00 max{𝝀𝑷𝒑}–min{𝝀𝑷𝒑} 0.00 0.00 1.00 0.00 𝝀𝑷     0.00 1.00 0.96 1.00 Alto Peru Cost Cost $22503 $24904 $24354 $29692 λC 1.00 0.67 0.74 0.00 Energy min{𝝀𝑬𝒑} 0.00 0.32 0.00 1.00 max{𝝀𝑬𝒑}–min{𝝀𝑬𝒑} 0.64 0.00 0.64 0.00 𝝀𝑬     0.17 0.32 0.20 1.00 Power min{𝝀𝑷𝒑} 0.00 1.00 1.00 1.00 max{𝝀𝑷𝒑}–min{𝝀𝑷𝒑} 1.00 0.00 0.00 0.00 𝝀𝑷     0.04 1.00 1.00 1.00  Cost. First, the cost (Cost) and the cost satisfaction (λC) are detailed. Note that the MILP essential demand solution has the minimum cost and the maximum satisfaction, while the MILP improved demand solution has the maximum cost and, therefore, the minimum satisfaction.  Energy and power. As the MILP model does not include satisfaction variables, they have been calculated manually. Moreover, the FMILP model for the minimum satisfaction assumption only provides the energy and power satisfaction for the least satisfied point (𝜆𝐸 and 𝜆𝑃, Section 3.2.1), so the satisfaction for each point is also calculated. Thus, the 24 table shows the minimum energy satisfaction of the least satisfied point (min{𝜆𝐸𝑝}), the difference between the most and the least satisfied points with regard to energy (max{𝜆𝐸𝑝}– min{𝜆𝐸𝑝}) and the average energy satisfaction of all points (𝜆𝐸     ). The same information is given for power. 5.3. Discussion about the case study application The results show that the MILP essential demand solution has a complete satisfaction for cost but low energy and power satisfaction. This means that this solution slightly exceeds the EpMIN and PpMIN values, due to small differences between the size of the generation equipment and users’ demand (in El Alumbre: 0.30 for the average energy satisfaction and 0.00 for power; in Alto Peru: 0.17 for energy and 0.04 for the average power satisfaction). The opposite situation is found for the MILP improved demand solution, where null satisfaction is achieved for cost and maximum satisfaction is attained for energy and power. As observed, these two solutions, which would have been obtained without the proposed FMILP models, represent extreme situations that do not properly balance satisfaction. In contrast, the FMILP model finds balanced solutions for the cost, energy and power satisfaction. For both case studies and both assumptions, the FMILP model finds a new solution (different from those found with the MILP model) that effectively represents a balance. The cost satisfaction is very high in these cases (from 0.60 up to 0.77), significant average energy satisfaction is achieved (from 0.20 up to 0.33) and very high average power satisfaction is obtained (1.00 or very close). In short, before the development of this work, rural electrification promoters had to manually test different energy and power values (for instance, an essential and an improved demands), and then discuss the results obtained in order to identify the most satisfactory solution. In contrast, the proposed fuzzy approaches allow to directly find a proper balance between the project cost and the electricity supply, hence increasing robustness of decision-making, as it happens in other works from the literature using fuzzy logic [Suganthi et al., 2015]. In addition, when comparing the two assumptions, the average satisfaction assumption allows compensations in the satisfaction between consumption points in exchange for cost savings. In contrast, the minimum satisfaction assumption obtains slightly more expensive solutions ($39799 vs. $38034 for El Alumbre and $24904 vs. $24354 for Alto Peru) but has better average energy satisfaction (0.33 vs. 0.30 for El Alumbre and 0.32 vs. 0.20 for Alto Peru); moreover, the least satisfied point regarding energy is more satisfied (0.32 vs. 0.00 in both communities) and the dispersion between consumption points is notably lower (0.13 vs. 1.00 for El Alumbre and 0.00 vs. 0.64 for Alto Peru). Indeed, it should be noted that in both communities the minimum satisfaction assumption achieves slightly higher average energy and power satisfaction values than the average satisfaction assumption, although at the cost of obtaining more expensive solutions. In short, the minimum satisfaction assumption prioritises proper energy distribution between end-users, while the average satisfaction assumption prioritises a lower cost for the two communities studied. These results allow the electrification project promoters to take better-informed decisions. 25 6. Conclusions The estimation of demand for populations accessing electricity for the first time is complex and subject to uncertainty. Project promoters must examine different demand values and study how the cost of the solution varies. In order to overcome such a limitation, this paper proposes to properly balance the energy and power supply vs the project cost. For this purpose, first, a deterministic MILP model is proposed, which minimises the project cost while simultaneously addressing the sizing and siting of PV-based systems, combining distribution networks and individual supplies. Then, as the inclusion of uncertainty in demand is not straightforward, five FMILP approaches are developed, considering different options to balance the satisfaction with regard to the cost, energy and power. The first three are based on the literature review, and balance the least satisfied issues vs the average satisfaction; while the other two proposed in this paper balance opposite satisfactions, cost vs supply (energy and power). In addition, each approach is developed for two assumptions: minimum satisfaction (which focuses on the least satisfied consumption point) and average satisfaction (which considers the global satisfaction of all consumption points). In order to validate the proposal, two computation experiments are performed. First, to compare the performance of the five FMILP approaches, considering 5 demand uncertainty scenarios and the 2 assumptions. The results show that literaturebased approaches do not capture the problem nature. In contrast, the new approaches properly balance the cost minimisation (tending to cheap and low-demand solutions) vs the energy and power maximisation (tending to expensive and high-demand solutions). In particular, the approach 4 obtains the highest performance and is a simple option, which directly compares the cost vs the energy and power demands without weighting nor calibration parameters, easing clarity in the analysis of results for the project promoters. A second computation experiment is carried out with two real communities from Peru in order to compare the results obtained with the deterministic MILP (that would be used if this work had not been developed) and the FMILP approach 4. The results show that the deterministic solutions represent extreme situations, while the fuzzy solutions achieve an overall good satisfaction for the cost and the demand. In short, an FMILP model is proposed, which can help promoters in developing countries to design electrification projects adapted to the particular characteristics of each target community, balancing the maximum energy and power supply vs the minimum project cost. The FMILP model has some limitations in terms of the technologies considered for electricity generation. Therefore, other renewable options, such as wind turbines, could be included in future research in order to cover a wider range of projects. In addition, the computation time for large instances might be challenging, so heuristic procedures could be developed to accelerate the solution of the problem. Acknowledgement This research was funded by the Spanish Ministry of Science and Innovation (RTI-2018-097962-B-I00) and the Centre for Cooperation Development of the Universitat Politècnica de Catalunya-BarcelonaTech. The authors are very grateful to the