Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints

Ouanes, Nesrine; Grandón, Tatiana González; Heitsch, Holger; Henrion, René Article — Published Version Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints Annals of Operations Research Provided in Cooperation with: Springer Nature Suggested Citation: Ouanes, Nesrine; Grandón, Tatiana González; Heitsch, Holger; Henrion, René (2024) : Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints, Annals of Operations Research, ISSN 1572-9338, Springer US, New York, NY, Vol. 344, Iss. 1, pp. 499-531, https://doi.org/10.1007/s10479-024-06287-9 This Version is available at: https://hdl.handle.net/10419/315294 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Annals of Operations Research (2025) 344:499–531 https://doi.org/10.1007/s10479-024-06287-9 ORIGINAL - OR MODELING/CASE STUDY Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints Nesrine Ouanes1·Tatiana González Grandón2·Holger Heitsch3·René Henrion3 Received: 28 February 2024 / Accepted: 10 September 2024 / Published online: 25 September 2024 © The Author(s) 2024 Abstract In this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In several rural areas in Africa with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: forecasting errors of solar power and load; frequency and outages duration from the main-grid. These uncertainties pose new challenges to the classicalpower system’soperationtasks.ThreealternativestotheJCCproblemarepresented. In particular, we present an Individual Chance Constraint (ICC), Expected-Value Model (EVM) and a so called regular model that ignores outages and forecasting uncertainties. The JCC model has the capability to guarantee a high probability of meeting the local demand throughout an outage event by keeping appropriate reserves for Diesel generation and battery discharge. In contrast, the easier to handle ICC model guarantees such probability only individually for different time steps, resulting in a much less robust dispatch. The even simpler EVM focuses solely on average values of random variables. We illustrate the four models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. The resultsshow the dispatchmodificationsforbatteryandDieselreserveplanning, with the JCC model providing the most robust results, albeit with a small increase in costs. Keywords Joint chance constraints ·Mini-grid operation ·Unreliable main grid ·Stochastic forecasting errors ·Spherical radial decomposition BNesrine Ouanes [email protected] Tatiana González Grandón [email protected] Holger Heitsch [email protected] René Henrion [email protected] 1Chair for Management Science, Humboldt University Berlin, Spandauer Straße 1, 10178 Berlin, Germany 2Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Sentralbygg 1, Gløshaugen Trondheim, Norway 3Weierstrass Institute Berlin, Mohrenstraße 39, 10117 Berlin, Germany 123 500 Annals of Operations Research (2025) 344:499–531 Index t∈TSet of time steps τSubset of time-steps when outages could start Parameters TTotal number of time steps c,tMarginal cost of Diesel at time t[$/kWh] cb,tMarginal aging cost of battery at time t[$/kWh] cg,tCost of day-ahead grid import at time t[$/kWh] pg,tPrice for day-ahead grid export at time t[$/kWh] cγ,tCost function for instantaneous grid exchange [$] pd,tPrice for electricity sales at time t[$/kWh] η+Average efficiency of battery charge [%] η−Average efficiency of battery discharge [%] σAverage battery self-discharge rate (internal resistance) [%] κAverage duration of outages [h] max Rated capacity of diesel [kW] bmax Maximum charging / discharging rate [kW] SOCmax Maximum state of charge of the battery [kWh] SOCmin Minimum state of charge of the battery [kWh] SOC0Initial state of charge of the battery [kWh] stForecasting of solar production at time t[kW] gmax Maximum available grid capacity at time t[kW] dtLoad forecasting at time t[kW] pProbability of successful islanding [-] ωProbability of an outage happening during one optimization horizon [-] Random variables δtLoad forecasting error at time t[kW] ξtForecasting error of solar power at time t[kW] γtSlack grid exchange at time t[kW] Non-negative decision variables tDiesel power at t[kW] g+ tElectricity imported from the main grid at time t[kW] g− tElectricity exported to the main grid at t[kW] b+ tBattery charge at time t[kW] b− tBattery discharge at time t[kW] SOCtStateofchargeattimet[kWh] rb tReserve for the battery discharge at time t[kW] r tReserve for the diesel at time t[kW] 1 Introduction The United Nations’ Sustainable Development Goal 7 (SDG-7) urgently calls for universal accesstocleanenergy, a mission that faceschallengesdespite increased electrification efforts. Projections by the international energy agency estimate that by 2030, around 650 million people might still lack access, and 9 out of 10 will be in West, Central, and East Africa 123 Annals of Operations Research (2025) 344:499–531 501 (IEA, 2020). Electrification via centralized grids is slow and expensive, prompting interest in decentralized mini-grid energy systems that offer quicker deployment and enhanced cost competitiveness (Antonanzas-Torres et al., 2021). Renewable-powered mini-grids (MGs) as defined by González Grandón et al. (2021)and Inensus (2014) are hybrid electricity supply systems combining wind turbine or photovoltaic (PV) generation (from 10 kW to 10 MW), energy storage systems, and (usually) a Diesel generator into low and medium voltage distribution networks. Classified by their relationship with the main power grid, MGs come in two forms: islanded mini-grids which operate independently, detached from the national (main) network and grid-connected mini-grids which link with the main grid and function as both backup systems for distribution as well as standalone units. This article focuses on the operational intricacies of grid-connected mini-grids, situated within the urban and peri-urban areas of Sub-Saharan African countries lacking in energy access. The paradox lies in the fact that, despite the presence of central-grid infrastructure, an alarming number of households, numbering in the hundreds of millions, still grapple with the challenge of accessing less than four hours of electricity per day (Rocky Mountain Institute, 2018). To counter this issue, governments in these countries are progressively adopting gridconnected mini-grids in areas with unreliable main grid supply. Yet, technical uncertainties inherent to grid-connected mini-grids in this context stem from multiple sources: stochastic solar power and demand forecast errors; absolute uncertain main grid outage onset times; and outage durations subjected to statistical analysis. As operational decisions are taken prior to the observation of the uncertainty, it becomes imperative to adopt suitable modeling methodologies that can effectively incorporate all these diverse forms of uncertainty. Thus, our primary research aim focuses on addressing the four mentioned uncertainties through modeling and algorithmic approaches, utilizing Joint Chance Constraints. Additionally, the study aims to assess the effectiveness of the proposed scheduling strategy by applying it to a case study example of a mini-grid in Lake Victoria. Introduced by Charnes and Cooper (1959), chance constraints offer an appealing tool for dealing with uncertainty in the constraints of an optimization problem. A classical and fundamental introduction to the theory and numerical treatment of chance constraints is presented by Prékopa (1995). A modern treatment is provided by Shapiro et al. (2014). Since theirintroduction,chance constraints have become common foreconomicdispatch problems, notably in hydro reservoir management (e.g., Berthold et al., 2022; Loiaciga, 1988; Prékopa & Szántai, 1978;vanAckooijetal.,2014), but also in power dispatch (e.g., Hong et al., 2022; Peña-Ordieres et al., 2021). For mini-grid or micro-grid dispatch, the approach has predominantly involved the use of purely-deterministic predictive models (González Grandón et al., 2021; Parisio et al., 2014). There is increasing interest, however, in applying probabilistic models in order to ensure sufficiently safe satisfaction of demand in a highly stochastic environment, not only with respect to renewable energy but also with respect to instabilities of the main grid. Zhao et al. (2014)andLiuetal.(2017) introduce the interesting concepts of probability of self-sufficiency and probability of successful islanding, respectively, in order to model the self-sufficiency of a mini-grid when isolated from the main grid by an outage. These concepts rely on keeping reserves for Diesel employment and battery discharge such that an outage of the main grid, to which the mini-grid is connected, can be survived based on these reserves with sufficiently high probability. The shortcoming of the models by Zhao et al. (2014)andLiuetal.(2017) is that they use individual (separate for each time in the given interval) chance constraints. While such a model is comfortable to handle since the chance constraint can be transformed without effort into an explicit equivalent, it does not really reflect the wish for robust self-sufficiency. Indeed, even if one may guarantee that 123 502 Annals of Operations Research (2025) 344:499–531 self-sufficiency holds true at each time individually with high probability, the probability of violatingself-sufficiencyatsome timemaybehighas well(or: theprobability ofguaranteeing self-sufficiency throughout a given period of time may be small). This is why we will rather consider joint chance constraints in this paper which are, however, more difficult to deal with. Indeed, they require the consideration of probabilities and their sensitivities with respect to the decision vector under multivariate random distributions. For numerical solutions related with joint chance constraints, we shall make use of the so-called spherical-radial decomposition of Gaussian random vectors which efficiently applies to Gaussian, Gaussian-like (e.g. multivariate log-normal, Gaussian mixture) or elliptically symmetric (e.g. multivariate Student) distributions and has found a lot of applications both in operations research and optimal control under PDEs with random coefficients (e.g., Berthold et al., 2022; Farshbaf-Shaker et al., 2020; González Grandón et al., 2017; Heitsch, 2020). The possibly striking difference between individual and joint chance constraints has been widely studied (e.g., Van Ackooij et al., 2010, p. 547); (Berthold et al., 2022,p.34). In this paper we focus on the challenges arising from the model with joint chance constraints. Therefore, we keep some other modeling aspects simple. In particular, we will not include binary decisions (simplified model for Diesel generator), we will not adequately model realistic battery ageing by means of differential equations and we will keep all decisions to be static with respect to the unfolding of uncertainty over time (thus ignoring the gain of information based on random observations prior to decision taking). The inclusion of all these aspects is subject of current and future work, e.g., by González Grandón et al. (2022). This article makes two key contributions, namely to the realm of applied joint chance constraint programming and to the advancement of SDG-7. More precisely, our analysis involves the following steps: •We base our input data on real power measurement data obtained from an operating MG in Tanzania and use forecasts for their solar PV generation and the electricity demand of the connected households. •We propose a model for an economic scheduling strategy of a MG connected to a weak main grid with a specified high reliability level. This model utilizes joint chance constraints, taking into account various uncertainties, including: 1) forecasting errors in renewable power generation, 2) forecasting errors in demand profiles, and uncertainties related to 3) frequency and 4) duration of outages. •We compare the proposed JCC model with three alternatives. We contrast results with an Individual Chance Constraint, Expected-Value Model and a (mainstream) so-called regular model that ignores outages and forecasting errors. 2 Mini-grid topology and modelling assumptions Figure1illustrates the grid-connected mini-grid configuration. This arrangement includes a solar PV generation unit, one battery energy-storage system (BESS), one Diesel genset, and one connection to an unreliable main-grid. The PV and BESS are integrated into the system via two distinct inverters linked to the Alternating Current (AC) bus. Interactions with the main grid occurs at the Point of Common Coupling (PCC). At the PCC, a transformer modifies the voltages, translating them from the lower level of the mini-grid to the higher-level of the primary grid (referred to hereafter as main grid or higher-level grid). In the examined context of this paper, this connection to 123 Annals of Operations Research (2025) 344:499–531 503 Fig. 1 Weakly Grid-connected Mini-grid Topology the main grid is unreliable, with outages occurring frequently due to failures in the main grid. This could be an overloading of the main grid, an uncurtailed excess generation or technical aspects due to failures of control strategies, physical components or transmission lines overheating. Lastly, the Diesel genset, acting as a supplementary component to bridge any electricity supply gaps, generates AC current directly, injecting it into the AC bus. Since the MG is balanced through an AC bus, both active and reactive powers will flow to the loads. In this paper, for simplicity purposes, we assume that the loads of all connected households maintain a purely resistive nature. We, therefore, conduct all our modelling using only active power flow figures, neglecting all reactive components. Further on, we neglect any temperature effects on the MG components and operation, especially for the battery storage system. The latter is modelled as one large synchronized unit that can either be charged or discharged, thus neglecting specific dynamics between the battery packs. Additionally, we model charging and discharging losses in the battery inverter as well as the battery’s internal resistance as single, average efficiency parameters. In terms of the flexibility of dispatch, we assume that the Diesel genset is able to run between0 and amaximalcapacity,withoutanyramping constraints orrestrictionson minimal resting and running times. Moreover, we follow the premise that the MG operator always maintains a continuous fuel supply. Additionally, we assume that the solar PV output can be controlled with the help of a Charge Controller (CC). In case of an excess of PV generation and an outage preventing any export to the main grid, the PV output can be curtailed fast enough such that the overall local power balance can be withheld. To model the demand-side, we aggregateallelectricityloadsintoone node, thus neglecting any line losses that occur at the local distribution level. Finally, we assume that the MG operatorhasaccessto time series forecasting of solar powergeneration and aggregatedemand fortheforthcoming24or48-hourtimeinterval. Furthermore, we assume that, whenan outage ofthemaingridoccurs,thepowerexchangeatthePCCdropsto0,thuspreventinganyplanned imports or exports. Outages can occur with a given probability at any arbitrary moment, the precise timing of which remains uncertain. We further assume that there is statistical evidence about the average duration of an outage from the main-grid (Klugman et al., 2021). Table 1 123 504 Annals of Operations Research (2025) 344:499–531 Table 1 Mini-grid uncertainties Source of uncertainty Assumptions Aggregate demand Access to a forecasting time series for the forthcoming 48h Solar PV generation Access to a forecasting time series for the forthcoming 48h Timing of main grid outages Unknown Duration of main grid outages Access to statistical evidence as an exogenous parameter (Klugman et al., 2021) summarizes the uncertainties present at the examined MG and assumptions made about their modeling. 3 Mathematical models This section defines the mathematical model used for the above described MG topology. As mentioned before, our aim is to find the optimal dispatch of a weakly-connected MG which allows one to survive a failure of the main grid by means of local energy supply (battery, Diesel). In order to achieve this goal, in addition to the planning of a regular dispatch for strongly connected MGs, reserves for Diesel and battery are included into the model. These can then be employed in the islanded mode of the MG in order to meet the local power demand. Since dispatch has to be planned prior to observing randomness in demand, solar radiation and outages, methods of stochastic optimization come into play. Inspired by the work of Zhao et al. (2014)andLiuetal.(2017), we find the application of probabilistic or chance constraints to be most appropriate here. They provide the possibility to find an optimal dispatch including reserves for Diesel and batteries such that the local demand in an islanded mode caused by an outage is met with a given probability (typically close to one). However, as mentioned in the introduction, these papers handle probability individually in time. While this approach eases drastically the computational treatment of chance constraints, it does not guarantee robust demand satisfaction throughout the whole critical time interval. Therefore, our main contribution is devoted to model the MG dispatch problem with respect to joint chance constraints. This type of constraints cannot directly be converted into explicit linear constraints as in the individual case, but requires to deal with probability functions and their derivatives under multivariate distributions. This section is organized as follows: after introducing the used nomenclature, we present the objective function and the (common) deterministic constraints for the optimization problem. Then, we introduce our additional joint chance constraints with respect to decisions on keeping reserves. We shall assume that an outage may occur at most once a day with a given probability. Its initial time is supposed to be uniformly distributed over the day. As for the duration of the outage, we either fix it (e.g., as a statistical mean from available historical data) or consider it to be an exterior parameter which can be varied in different computations. Next, we move on to a model with individual chance constraints as used by Zhao et al. (2014) and Liu et al. (2017), in order to illustrate the gain in robustness by using joint chance constraints. Finally, for the sake of completeness, we also mention two further simplifications of the model: first, the expected-value model in which we replace all random parameters by 123 Annals of Operations Research (2025) 344:499–531 505 their expected values, such that statistical information is reduced to the first moment. Second, the widely-used regular model, which in addition to the previous simplification, also completely ignores the possibility of outages (Beath et al., 2023; Elegeonye et al., 2023; González Grandón et al., 2021;Kumar,2021; Kumar and Pahuja, 2021). 3.1 Objective functions In this subsection, we will introduce the objective function used across the mathematical models starting with the cases of a strongly-connected and a weakly-connected mini-grid. Strongly-connected mini-grid We first present the objective function for a strongly-connected mini-grid, i.e., where outages do not occur. In such a mini-grid, if the operator faces forecasting errors in solar and load predictions, the impact is negligible from a physical standpoint. The system can be adapted by managing exports and imports. However, economically, it’s important to note that instantaneous grid exchanges come with a higher cost compared to day-ahead exchanges. Forsuchaproblem,wewanttomaximizetheprofitoftheMGdispatch,which isequivalent to minimizing the net operational costs minus net revenue. The summation term in the objective (1) reflects the negative profit of the MG over the nominal time horizon T,plusthe duration of an outage κ, as the outage could start at the end of the nominal time horizon. f0(, b+,b−,g+,g−):= T+κ  t=1 c,t·t+cb,t·(b+ t+b− t)−pd,t·dt+cg,tg+ t−pg,tg− t+E[cγ,t(γt)].(1) The aim is to maximize the profit of the MG dispatch, i.e., minimize the function f0,where revenues from electricity sales are subtracted from the marginal costs of operation. The marginal costs consist of the fuel costs c,tfor the Diesel genset, of the battery lifetime depreciation costs cb,tdue to the discharge-charge cycling, and of the costs cg,tfor importing from the main grid. Solar power is assumed to have zero marginal costs and therefore doesn’t appear in the objective function. Thus far, maintenance costs are not explicitly considered. They could be levelized and included in the cost coefficients of each technology. However, for this paper, further considerations about the maintenance approach are omitted. The revenues are generated from selling the electricity at exogenous price of pd,tto the connected households consuming electricity at the MG level, as well as from exporting excess electricity at the exogenous price of pg,tto the main grid. The random variable γt=γt(z,ξ,δ):= dt+δt−st−ξt−t−b− t+b+ t−g+ t+g− t,∀t=1,...,T+κ, with z:= (, b−,b+,g−,g+), (2) isdefinedasthecompensatingactionestablishingtheloadbalanceafterapplyingthedecisions zon the employment of Diesel t, battery btand grid exchange gton a day-ahead approach and observing the random variables solar energy st+ξtand demand dt+δt(given forecast plus random deviation in both cases). A positive value of γtrepresents a lack of energy to be compensated by instantaneous import from the main grid (at a price that may be substantially higher than for the import on the day ahead market). A negative value of γtrefers to an excess of energy which has to be instantaneously exported to the grid, again at certain costs (so that, 123 506 Annals of Operations Research (2025) 344:499–531 contrary to the day ahead market there is no reward for the export). As γtand, hence, its associated costs, are random, we minimize the expectation of these costs. All costs and prices are defined as time-variable parameters, hence the index tfor each of these parameters. Weakly-connected mini-grid Here, we introduce the intricacies associated with a weakly-connected MG, i.e. with potential outages in the main grid and adjust the objective function accordingly. Starting with the initial objective function (1), we adapt the objective function f0to account for the case of possible grid outages. We introduce the following definition: f1(, b+,b−,g+,g−,r,rb):= T+κ  t=1 c,t·t+cb,t·(b+ t+b− t)−pd,t·dt +ω T T  τ=1T+κ t=1,t=τ,...τ+κcg,tg+ t−pg,tg− t+E[cγ,t(γt(z,ξ,δ))]+ τ+κ  t=τ c,tr t+cb,trb− t +(1−ω) T+κ  t=1cg,tg+ t−pg,tg− t+E[cγ,t(γt(z,ξ,δ))].(3) The first term is defined in analogy to (1) including again a summation over the Diesel and batterycyclingcostsminus the revenuesgeneratedbysellingtheelectricity to local customers at a given price. Now, considering the possibility of grid outages, we delineate two scenarios for the costs and revenues associated with grid exchange: one when an outage occurs and another when it does not. These scenarios are articulated in the second and third terms of (3). We make the assumption that, at most, one outage may occur during the designated day ahead, and the probability of its occurrence is denoted as ω. In the absence of an outage during the specified day (probability equals 1 −ω), the sole additional costs stem from trading with the main grid (third term). Conversely, when an outage occurs, suppose at time τ,thenin the interval [τ,...,τ +κ], representing the duration of the outage, no trading costs with the main grid are incurred (due to the lost connection). However, additional costs arise from utilizing reserve capacities r t,rb− tfor Diesel generation and battery discharge to meet the local demand in the islanded mode (second term). We assume that τis uniformly distributed, indicating that an outage may initiate at any time with equal probability. Consequently, in the second term of (3), we compute the average costs across all potential starting times for the outage. In contrast, the duration κof the outage is regarded as the statistical mean derived from historical data or treated as an exogenous parameter of the problem, capable of assuming multiple values. 3.2 Deterministic constraints The physical constraints represent well-known formulations for mini-grid economic dispatch models, consisting in capacity constraints of the MG components, a power balancing constraint for MG supply and demand, inter-temporal constraints for the storage state of charge, as well as considerations of conversion and distribution losses expressed through average efficiency parameters. These will be discussed in more detail in the following. Capacity constraints of the Diesel genset: it is assumed that the Diesel genset can operate between 0 and its maximal capacity max. Therefore, the sum of the decision variables for 123 Annals of Operations Research (2025) 344:499–531 513 Expected-value constraints ignoring forecasting errors Next, we further simplify the model by maintaining the assumption of a weakly-connected mini-grid, while completely avoiding probabilistic constraints. Instead, we require that the constraintsforsuccessfulislandingholdonaverage.Specifically,thisexpected-value dispatch model does consider outages but only considers the expected value of forecast errors for both solar production and for demand in (20). As a result, the system of constraints describing successful islanding is given by: dt−st−t−b− t+b+ t≤rb t+r t,∀t=1,...,T+κ. (24) Observe that (24) represents the same linear inequality system as (23) but with different right-hand side. We note here that we choose to define the system of constraints (23)and(24)fort= 1,...,T+κso that we can apply the objective function defined in (3) across all models that consider outages, thus allowing for a comparability between the results of the simplified models (ICC and expected-value models) and the JCC model. Regular dispatch ignoring outages and forecasting errors The final model we introduce is the simplest and most basic, as it ignores both forecasting errors and the occurrence of outages from the main-grid. We refer to this as the regular model due to its widespread use in the literature, where the operation or economic dispatch of minigrids often overlooks uncertainties (Beath et al., 2023; Elegeonye et al., 2023; González Grandón et al., 2021;Kumar,2021; Kumar and Pahuja, 2021). These models typically adopt a deterministic approach, focusing solely on the expected value of the forecasting time series, disregarding both their random errors and the uncertainty of timing of blackouts. The first simplification in this model is the elimination of the last term in the objective function f0 presented in (3), which accounts for the random instantaneous exchange with the grid. The second simplification concerns the assumption of a strongly connected mini-grid, where outages are disregarded. Consequently, the reserve variables for Diesel and battery systems, which are included in other models to ensure the reliability of supply during blackouts, are also excluded from this model. Under these simplifications, we obtain the following minimization objective function: f2(, b+,b−,g+,g−)= T+κ  t=1c,t·t+cb,t·(b+ t+b− t)−pd,t·dt+cg,tg+ t−pg,tg− t.(25) We also adjust the power balancing constraint from (9) as follows: st+t+b− t−b+ t+g+ t−g− t,=dt∀t=1,...,T+κ. (26) Furthermore, since the regular model doesn’t include any reserve considerations, we adjust the box deterministic constraints for the Diesel and battery discharge variables as shown in (27)-(28), and remove the constraint for the potential state of charge presented in (13). Meanwhile, the constraints (7), (8)and(10)-(12) remain unchanged. Capacity constraints of the Diesel genset 0≤t≤max,∀t=1,...,T+κ. (27) 123 514 Annals of Operations Research (2025) 344:499–531 Capacity constraints of the battery discharge 0≤b− t≤bmax,∀t=1,...,T+κ. (28) 4 Case study Our case study focuses on an operational mini-grid in Chifule, an island of Lake Victoria, Tanzania. Currently, the MG remains isolated, lacking any linkage to the main-grid. However, considering the research question at hand and anticipating a future where many MGs in subSaharan Africa will be connected to weakly-connected main grids, we assume that the MG, for the purposes of this study, is connected to a main grid characterized by unreliability and frequent outages. In the following, we present the physical and monetary parameters preset for this case study. For the random parameters used in this case study, we refer to Sect.3.3 where we derive the parameters characterizing the random components of our model, i.e., the mean and (co)-variances of the distribution functions resulting from the presented forecasting models. Physical parameters The physical parameters pertinent to this context are detailed in Table 2, while the price and cost parameters are presented in Table 3. In conjunction with the parameters listed in these tables, it is assumed that the nominal time horizon is 24h, and the time step is set at one hour. All parameters, excluding those pertaining to the main grid connection, are defined based on data from the MG operator in Tanzania. For the parameters related to the grid connection, we use heuristic and model-informed values. For instance, for the maximum available grid capacity gmax, we assume a high value meaning that in case of grid availability, the grid can always guarantee the satisfaction of the local electricity demand. For the average duration of an outage κ, we initially set its value to 3h, based on empirical values collected from main-grids in similar geographies (Klugman et al., 2021). The κparameter will be varied, further on, to examine its effect on the obtained solutions. For the probability of an outage ω, we assume that outages occur on nine days out of ten, i.e., ω=0.9 to magnify the effect of outages on the daily planning of the dispatch. Monetary parameters The monetary parameters, presented in Table 3are also based on indications by the MG operator for the marginal costs of the fuel, the electricity sale price to local customers, and scalar aging costs. The marginal aging cost of the battery is the monetary value reflecting the cost of battery aging due to an additional power unit charged or discharged from the battery Xu et al. (2018). For the cost of importing from the grid and the price of exporting to it, we use values derived from expert interviews with weakly-connected MGs in sub-Saharan Africa. Thereby, we assume that prices and costs differ between night and day, reflecting market conditions, i.e., that importing from the grid costs more during the night than during the day, since at night, the main grid is overall more stressed due to the absence of (cheap) renewable-energy generation (e.g., solar), and that selling to the grid during the day is more valuable due to the higher demand. 123 Annals of Operations Research (2025) 344:499–531 515 Table 2 Physical parameters Parameter Unit Value Description Battery system η+% 95 Average battery charging efficiency η−% 95 Average battery discharging Efficiency σ% 0 Battery self-discharge rate bmax kW 10 Maximal battery charging/disCharging rate bcap kWh 100 Maximal battery capacity SOCmax % 90 Maximal battery state of charge SOCmin % 20 Minimal battery state of charge SOC(t=0)% 35 Initial battery state of charge Diesel genset max kW 5 Rated Diesel genset capacity Main grid connection gmax kW 100 Maximum available grid capacity κ3 Average duration of outages ω0.9 Probability of an outage occurring during one day Solar input stkW see Fig.3Forecasting of maximum solar Production Meta-parameters p% 90 Probability of Successful islanding dtkW see Fig.2Forecasted electricity load Table 3 Cost & price parameters Parameter Unit Values Description c,te/kWh 0.35 Marginal cost of Diesel cb,te/kWh 0.0055 Marginal cost of battery cg,te/kWh 0.55 (night), 0.15 (day) Marginal cost of grid import cγ,te/kWh 0.85 (night), 0.45 (day) Cost for instantaneous grid exchange pg,te/kWh 0.08 (night), 0.13 (day) Price for grid export pd,te/kWh 0.55 Price for electricity sales to customers 5 Results In this section, we present the results of the optimization problem of the regular (purelydeterministic), expected, ICC and JCC models and compare them to investigate the effects achieved due to the different formulations and ways of handling the uncertainties. For these results, we use the previously presented input parameters and time series. Optimal dispatch First, we look into figures 4-7showcasing the optimal dispatch of each component (respectively-colored bars), in order to meet the forecasted electricity demand profile (as 123 516 Annals of Operations Research (2025) 344:499–531 Fig. 4 Economic dispatch in the regular model. Arrows near the x-axis show an increase or a decrease in the price/cost of grid export/import a continuous black line) given the forecasted solar power production (as a continuous yellow line). To emphasize the price signaling that is implied by the chosen monetary parameters for grid exchange (see Table 3), we show using top and down arrows near the x-axis of the figures the changes in grid import cost (dark blue) and grid export price (light blue). A top arrow indicates an increase in the cost parameter, while a down arrow indicates a decrease. In this sense, the cost of grid import drops to its set lower level at 9 a.m. and increases again to its set higher level at 6 p.m. each day, while the price for grid export increases at 9 a.m. and drops at 10 p.m. The dispatch of the regular model is shown in Fig.4. The figure shows that the demand is mainly met through discharging the battery for the first two morning hours and from 10 p.m. to midnight. In the early morning hours starting from 3 a.m., we notice an employment of the Diesel genset. Later, starting from 9 a.m., as the prices for main grid electricity import get cheaper (see Table 3), electricity is also imported from the main grid for a few hours. During the day, the demand is met by using the solar PV generation with a peak around 3 p.m. The solar PV power is first used to meet the electricity demand directly. Any remaining generation from the solar system is either sold to the main grid for additional revenue generation or used tochargethebatteryin preparationforthe nighthours.This optimaldispatch leadstoexpected profits of 84.46 efor the optimization horizon T+κ, i.e., 27h. Looking into the expected-value model, where the model ignores forecasting errors but factor in the possibility of main grid outages occurring with a certain probability, the resulting dispatch shown in Fig.5differs from the regular model dispatch, mainly by using the Diesel genset more strongly (throughout the first eight hours of the day) and using more of the excess solar PV generation to charge the battery in the middle of the day. Since the increased Diesel employment adds to the operational costs and the decreased exporting to the main grid reduces the generated profits, the overall expected profits drops to 69.06 efor the optimization horizon of 27h. 123 Annals of Operations Research (2025) 344:499–531 517 Fig. 5 Economic dispatch for the expected-value model. Arrows near the x-axis show an increase or a decrease in the price/cost of grid export/import The expected optimal profits drop even more for the ICC model, where we use individual probabilistic constraints to ensure a certain probability of successful islanding while considering both main grid outages and forecasting errors. For a probability of 90%, the optimal profits are 63.73 e. In the resulting optimal dispatch shown in Fig.6, the optimizer hedges against forecasting errors and main grid outages by starting to charge the battery sooner (i.e., at 1, 2, 5 and 6 a.m.) with electricity imported from the main grid. Additionally, more of the excess electricity is used to charge the battery than to export to the main grid and more Diesel generation occurs during the night hours in order to reduce the electricity discharged from the battery keeping it as a reserve. This new dispatch clearly leads to higher Diesel and main grid import costs and lower main grid export revenues. Overall, compared to the expected-value model, the profits with the ICC model drop by merely 5 e, all while ensuring a high probability in the order of 90% individually at each time step to successfully meet the local demand in an islanded mode and compensate for forecasting errors. Still, as explained in earlier sections, the ICC model guarantees the required level ofislanding probability only for the single time step when an outage starts and doesn’t cover the entire duration of the outage. For this, we look into the dispatch results of the JCC model, shown in Fig.7. The figure showcases the same effects as for the ICC model, only larger in scale, i.e., the dispatch shows more battery charging through main grid imports during the night and excess solar PV power during the day, as well as less battery discharging during the night hours replaced by Diesel generation. As a consequence, the expected profits drop to 59.87 e, yet with a guarantee that the minigrid survives islanding and probable forecasting errors with a probability of p=90% for the whole duration κ=3 of the main grid outage. Further in this section, we analyze the influence of the parameters pand κon the expected profits and achievable successful islanding probabilities. One concluding remark regarding the optimal dispatch shown in the figures above, is that, except for the regular model which doesn’t include any random variables, the planned dispatch from the MG components do not balance out to the forecasted demand given the 123 518 Annals of Operations Research (2025) 344:499–531 Fig. 6 Economic dispatch for the ICC model. Arrows near the x-axis show an increase or a decrease in the price/cost of grid export/import Fig. 7 Economic dispatch for the JCC model. Arrows near the x-axis show an increase or a decrease in the price/cost of grid export/import forecasted solar power generation. This can be explained by the definition of the balancing constraint as shown in (9). With this equation, we don’t impose the expensive condition that forecasted demand be met with the respective dispatch from the MG components. Instead, we use for the power balance the real demand and the real solar power generation, as well as the variable γtas a random variable to cover the instantaneous grid exchange (see (2)and the linked explanation in that section of the paper). To showcase this effect in a clearer way, we present, in Fig.8, the net planned dispatch from all the components in the JCC model in each time step, the forecasted demand and one realization of the real demand. In addition, we showcase the instantaneous dispatch, composed by one realization of the solar power 123 Annals of Operations Research (2025) 344:499–531 519 Fig. 8 Planned and instantaneous dispatch of MG components, and forecasted and real demand in the JCC model with cγ,t=0.85e/kWh (night) and 0.45 e/kWh (day) forecasting error ξand the random grid exchange γ. The figure shows clearly the deviations between the planned dispatch and the forecasted demand, which are balanced out with the instantaneous dispatch, e.g, at time steps 5 and 7 as a negative deviation from the forecasted demand translating into an instantaneous need for more grid import, and in most other time steps, e.g., at times 9 to 24 as a positive deviation, translating into an instantaneous need for more grid export or generation curtailment. An immediate result of this modeling logic is that the amount of power that is balanced out using the instantaneous grid component γtvaries depending on the predetermined cost of γt, denoted as cγ,t: when the the cost associated to the instantaneous grid exchange cγ,t and modeled in the objective function using E[cγ,t(γt)]is higher, less power is balanced using the expensive, instantaneous grid exchange γand the planned dispatch matches the forecasted demand more. In Appendix B, we showcase the effect of varying cγ,ton the resulting dispatch. Optimal battery cycle Thepresented differencesindispatch can also beexaminedintermsof optimal batterycycling strategy, shown in Fig.9-12 as charging and discharging powers (in different shades of green) and SOC development (as a continuous black line). Compared to the cases of the regular and expected-value models, the chance constraint models showcase a strategy to charge the battery overnight, which is understandably more significantinthecaseoftheJCCmodel.Consistently,thechanceconstraintsmodelsshowcase less discharging of the battery in the night hours, this being at the lowest for the JCC model. Looking closer at the range of battery cycling, we note that for the regular and the expectedvalue model, the state of charge is, as expected, cycled to lower levels than for the ICC and JCC models. In the former two, the state of charge reaches its preset minimum value of 20 %, while, for the ICC, it only goes slightly below 30% and, for the JCC, it remains slightly above 30%. 123 520 Annals of Operations Research (2025) 344:499–531 Fig. 9 Battery cycle for the regular model Fig. 10 Battery cycle for the expected-value model Optimal reserves Finally, we take a look at the reserves planned for the entire optimization horizon. Starting with the reserves in the case of the expected-value model shown in Fig.13, we note that the planned diesel reserves cover almost entirely the planned grid import (see Fig5), thus ensuring that the minigrid succeeds in its islanding provided that there is an outage during these hours that would hinder the import of power. Fig.14 presents the planned optimal reserves for the case of an ICC model. Again here, we note that the reserves are planned at each time step of planned grid import, as shown in Fig.6. During the peak of solar generation in the hours 12-19, where excess electricity is recorded, no reserves are planned. 123 Annals of Operations Research (2025) 344:499–531 521 Fig. 11 Battery cycle for the ICC model Fig. 12 Battery cycle for the JCC model For the case of a JCC model, Fig.15 shows that more battery and Diesel reserves are planned in all time steps and even in more hours of the day than in the ICC case, i.e., only in time steps 13-18, no reserves are present. The planned reserves translate into additional possible profits that are held back, e.g., through avoided grid export revenues or increased Diesel deployment costs, thus explaining the gradual worsening of the objective function value from the regular to the JCC case (see Table 4). However, in the following, we prove the increased reliability of the MG operation that is achieved by the JCC model, justifying the small drop in profitability showcased so far. 123 522 Annals of Operations Research (2025) 344:499–531 Fig. 13 Reserves for the expected-value dispatch model Fig. 14 Reserves for the ICC dispatch model Increased reliability With the ICC and JCC model, we are capable of handling many uncertainties in the MG operation,as explainedin Sect.2,thus increasing the reliabilityofmeetingthelocal electricity demand. Thereby, we handle different types of uncertainty: on the one hand, the uncertainties about the forecasting of the solar PV generation and the electricity demand, on the other hand, the uncertainties about an outage of the main grid. The uncertainties about when and how long a main grid outage could occur enlarges the uncertainties about the forecasting errors: provided that the grid is reliable and doesn’t experience any outages during one day, the main grid could level up any forecasting errors and ensure that the power balance is always met. Nevertheless, in the presence of an unreliable grid, not only are forecasting errors 123 Annals of Operations Research (2025) 344:499–531 529 Fig. 20 Planned and instantaneous dispatch of MG components, and forecasted and real demand in the JCC model with cγ,t=0.65 e/kWh (night) and 0.25 e/kWh (day) Fig. 21 Planned and instantaneous dispatch of MG components, and forecasted and real demand in the JCC model with cγ,t=1.35 e/kWh (night) and 0.95 e/kWh (day) Acknowledgements We thank INENSUS GmbH for their commitment to open science by sharing real power measurement data from one of their mini-grid subsidiaries in Tanzania. The second author is thankful for the support by the Fondation Mathématique Jacques Hadamard (FMJH) Program Gaspard Monge in optimization and operations research including support to this program by Électricité De France (EDF) No P-2022-0022. The third author is thankful for the support by the Deutsche Forschungsgemeinschaft (DFG) in the Collaborative Research Centre CRC/Transregio 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, Project B04. The fourth author acknowledges the support by the DFG ExC 2046 MATH+: Berlin Mathematics Research Center under project AA4-10. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest The authors declare that they have no known competing financial or non-financial interests that could have appeared to influence the work reported in this paper. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory 123 530 Annals of Operations Research (2025) 344:499–531 regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References Antonanzas-Torres, F., Antonanzas, J., & Blanco-Fernandez, J. (2021). 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