Disentangling Sources of Future Uncertainties for Water Management in Sub-Saharan River Basins

Water management in sub-Saharan African river basins is challenged by uncertain future climatic, social and economical patterns, potentially causing diverging water demands and availability, as well as by multi-stakeholder dynamics, resulting in evolving conflicts and tradeoffs. In such contexts, a better understanding of the sensitivity of water management to the different sources of uncertainty can support policy makers in identifying robust water supply policies balancing optimality and low vulnerability against likely adverse future conditions. This paper contributes an integrated decision-analytic 5 framework combining optimization, robustness, sensitivity and uncertainty analysis to retrieve the main sources of vulnerability to optimal and robust reservoir operating policies across multi-dimensional objective spaces. We demonstrate our approach onto the lower Umbeluzi river basin, Mozambique, an archetypal example of sub-Saharan river basin, where surface water scarcity compounded by substantial climatic variability, uncontrolled urbanization rate, and agricultural expansion are hampering the Pequenos Lipompos dam ability of supplying the agricultural, energy and urban sectors. We adopt an Evolutionary 10 Multi-Objective Direct Policy Search optimization approach for designing optimal operating policies, whose robustness against social, agricultural, infrastructural and climatic uncertainties is assessed via robustness analysis. We then implement the GLUE and PAWN uncertainty and sensitivity analysis methods for disentangling the main challenges to the sustainability of the operating policies and quantifying their impacts on the urban, agricultural and energy sectors. Numerical results highlight the importance of robustness analysis when dealing with uncertain scenarios, with optimal-non robust reservoir operating policies 15 largely dominated by robust control strategies across all stakeholders. Furthermore, while robust policies are usually vulnerable only to hydrological perturbations and are able to sustain the majority of population growth and agricultural expansion scenarios, non-robust policies are sensitive also to social and agricultural changes, and require structural interventions to ensure stable supply.

in order to ensure continuity in urban supply. This dry pattern is expected to be further exacerbated by climate change in the coming years, with an estimated precipitation decrease of about 10% and a temperature increase of 3 • C (Fryslân et al., 2014).
To mitigate the effect of frequent and prolonged drought episodes hitting southern Mozambique, the World Bank have recently financed the Greater Maputo Water Supply Expansion project (GMWSEP, Miguel (2019)). The project consists in a set of infrastructural interventions, mainly constituted by a water treatment plant downstream the Corumana dam, in the 95 adjacent Sabie river basin, and by a 95 km pipeline connecting Corumana with the city of Maputo, ensuring an additional water supply capacity of q C of 1.8 m 3 s −1 (about 70% of the current urban water demand). The GMWSEP is expected to play a key role not only in mitigating drought effects but also in sustaining the rapidly increasing population of Maputo.

Umbeluzi model
We model the Umbeluzi river system (Fig. 1b) using a combination of conceptual and empirical models assuming a daily time where e t+1 is net inflow to the reservoir in the interval [t; t + 1) (i.e., inflow minus evaporation and seepage losses) and r t+1 is the volume released in the same interval. This is further decomposed into r d t+1 and r up t+1 , representing the downstream releases 105 for hydropower production, urban and irrigation supply, and the upstream pumping from the upstream irrigation districts, respectively. The actual releases r d t+1 = f 1 (s t , u d t , u up t , e t+1 , t) and r up t+1 = f 2 (s t , u up t , u d t , e t+1 , t) are formulated according to the nonlinear, stochastic relations f 1 (·) and f 2 (·) between r up t+1 , r d t+1 and the release decisions u up t and u d t (Piccardi and Soncini-Sessa, 1991). The latter are in fact constrained by physical constraints (i.e. spillway activation and inactive storage threshold) within a discretionary operating space by the maximum and minimum feasible release function (see Soncini-Sessa 110 et al. 2007 for more details). Such constraints, in turns, implies interdependence among the release decisions, so that the total release never exceeds the minimum feasible release. We assume here feedback operating rules, parametric in ζ, where decisions are conditioned upon the current system conditions i.e. where I t represents the information upon which the policy is based. The analytical expression for the functions f and g depends upon the optimization problem formulation, and it is provided in section 3.2.2.
The aggregated downstream irrigation supply is modelled by means of a diversion dam, represented mathematically with an empirical exponential function (Celeste and Billib, 2009) parametric in α and β of the form: where W d t is the aggregated downstream irrigation demand ( Figure 2d). The periodic sequence of the control laws (over a period of 365 days) described in equations 2 and 3 constitutes the control policy π θ , where θ = |ζ, α, β| represents the vector of the control policy parameters.
The diversion rules allow hedging the water abstractions to account for downstream users (i.e. the city of Maputo), and the 125 actual diverted flows r i t+1 are constrained by the the environmental flow requirement in the Espirito Santo Bay q e t as follows.
where q max is the maximum divrsion channel capacity.
Finally, the amount of water to be diverted for urban supply to the city of Maputo r w t+1 is computed as: where W w t is the urban demand. The data necessary for the implementation of the analysis for the time-period December 2016-January 2006 were provided by The Administracão Regional de Ãgua (ARA) Sul, which is the water agency responsible for river basin management in southern Mozambique (including the Umbeluzi).

Integrated Decision-Analytic Framework
The integrated decision-analytic framework we adopted is represented in Figure 3. The approach is composed by three main blocks: optimization (O), robustness (R), and sensitivity-uncertainty (SU).
Block O is responsible for generating operating policies which are optimal under historical conditions. Block R extracts the optimal policies that are also robust against future changes in climate (climatic uncertainty), irrigation demand (irriga-140 tion demand uncertainty), infrastructures (infrastructural uncertainty), and urban demand (population uncertainty); block SU bridges robust policies (RP) with the operating objectives variability in response to uncertain input realizations, and allows the identification of main the sources of vulnerability for the hydrosystem.
The three blocks are interconnected as follows: first, to explore tradeoffs among stakeholders under the historical climatic, agricultural, infrastructural and urban demand drivers value (i.e., the baseline), we define the operating objectives, and run a 145 6 https://doi.org/10.5194/hess-2021-40 Preprint. Discussion started: 26 January 2021 c Author(s) 2021. CC BY 4.0 License. multi-objective evolutionary algorithm to identify the optimal operating policies via optimization-based simulation. Then, we assess their robustness with respect to the future evolution of the drivers by perturbing the baseline across all the four (climatic, agricultural, infrastructural and population) uncertainty dimensions to generate the states of the world (SOW). We then resimulate the system iteratively perturbed by the SOW for each of the optimal operating policies identified via optimization and compute the worst objective function values. The robust policy for each stakeholder is subsequently identified as that 150 yielding the best performances in the worst condition (minimax robustness metric). To disentangle the sources of vulnerability for RP, we implemented an uncertainty and sensitivity analysis framework. In particular, uncertainty analysis is employed to quantify the objective function variability in response to the uncertain future evolution of the system's drivers. Here, following the well known definition of behavioral parameters (Beven and Binley, 1992;Montanari, 2005), we identify the behavioral system perturbations as those SOWs satisfying predefined performance requirements, i.e. those yielding to acceptable objective 155 function values. Sensitivity analysis is responsible for determining the relative contribution of each individual uncertainty source in shaping the objective space and for ranking them across policies and objectives by means of a sensitivity index.  Further details about each block in Figure 3 are provided in the following sections. We model the stakeholder's affected by the operation of the BPL dam through the following four utility functions: 1. Upstream irrigation deficit J IU , expressed as the square difference between water supply and demand (to be minimized): Where N (days) is the simulation horizon, N y are the number of years in the simulation horizon, and b t is a weight 165 representing higher losses when the deficit occurs during the growing season. W up t and r up t+1 are the irrigation demand and the amount of water pumped from the reservoir to upstream irrigation, respectively.
2. Downstream irrigation deficit J ID defined similarly as in equation 6 (to be minimized): where W d t and r i t+1 are the irrigation demand and the reservoir releases diverted for downstream irrigation, respectively. 3. Urban deficit for the city of Maputo J U D , computed as the difference between urban supply and demand (to be minimized): where W w t , r w t+1 and q C t are the urban demand, the reservoir release diverted for urban supply and the additional inflow from Corumana, respectively. 175 4. Hydropower production in the BPL power plant J HP (to be maximized): where HP t is the hydropower production on day t, defined as: where η is the turbine efficiency, g = 9.81 m/s 2 is the gravitational acceleration, γ w = 1000 kg/m 3 is the water density, 180 h t is the net hydraulic head and r d t is the turbined flow.

EMODPS
The optimal operating policies under the baseline are designed using Evolutionary Multi-Objective Direct Policy Search (EMODPS) (Giuliani et al., 2016b). EMODPS is a simulation-based optimization approach, which has been recently demonstrated (Giuliani et al., 2016b) to successfully overcome the major limitations associated with traditional Stochastic Dynamic DPS is employed to explore the parameter space θ = |ζ, α, β| of the system operating policy π θ that optimizes the expected long term cost, i.e.: where: subject to equations 1 to 5 where finding π * θ means finding: (1) the optimal parameters ζ * ∈ Z of the BPL reservoir operating policy; and (2) the optimal 195 parameters [α * , β * ] ∈ Θ irr for the regulation of the irrigation diversion canal. The parameters are intended optimal with respect to the objectives J θ . The reservoir operating policy is selected such that policy inputs I t can provide information feedback for the upstream and downstream release decisions u up t (I t ), u d t (I t ). In this study, it is represented with a nonlinear approximating network of the Gaussian radial basis function family (RBFs), which are known for well-validating out of sample data (Giuliani et al., 2016b;Quinn et al., 2019). 200 Mathematically, the operating policy can be expressed as: Where n is the number of RBFs, w k i is the weight of each RBF, m is the number of inputs, and c and b are the center and radii of the RBF. The reservoir operating policy parameter vector is therefore constituted as: , and the number of parameters n ζ to be found is n(2m + k).
In order to explore the parameter space and discover optimal values, we employ Multi-Objective Evolutionary Algorithms.
The term evolutionary refers to the natural randomized mating, selection, and mutation processes that are mimed by the algorithms to evolve a Pareto-approximate set of solutions (Deb, 2001;Coello et al., 2007). MOEAs have proved to successfully deal with complex multiobjective optimization problems, including water reservoir operations (Maier et al., 2014). In this 215 study, we employ the self-adaptive Borg MOEA (Hadka and Reed, 2013), which has been shown to guarantee high robustness in solving a variety of multiobjective problems when compared to other MOEAs (Salazar et al., 2016).

Robustness
We perform a robustness analysis with the aim of evaluating the robustness of the various operating policies identified via EMODPS over an ensemble of future realizations of the climatic, agricultural, infrastructural and urban demand drivers. Ac-220 cording to Herman et al. (2015), a robustness analysis is usually carried out by performing the following sequential steps: (1) generation of alternative policies; (2) sampling of possible future scenarios and; (3) computation of robustness metric. Here, the alternative operating policies are those identified via optimization, while the generation of future scenarios is performed by perturbing historical trajectories assuming independent uniform distribution for the perturbation multipliers . The multiplier range is either defined by a-priory expert knowledge of the system or based on experimental 225 results. Climatic, agricultural, infrastructural and population scenarios are then combined to generate a set of uncertain states of the world. Finally, among the robustness metrics available in the literature (for a review see McPhail et al., 2018), we select minimax, a metrics that identifies the solution providing the best performance assuming the realization of the worst conditions.
More details on the formulation of states of the world and on the robustness metrics are provided below.
They are driven by uncertainty in (1) streamflow, (2) irrigation demand, (3) additional inflow from Corumana dam, and (4) population growth rate, and are here characterized as follows: 1. Climatic uncertainty: We generate high resolution scenarios of rainfall and temperature for the Umbeluzi river basin to the year 2100 by the quantile-quantile mapping (QQ-Mapping) downscaling procedure. We apply QQ-Mapping to 235 coarse resolution data from three different Regional Circulation Models (ICHEC RCA4, ICHEC RACMO and ICHEC HIRHAM5, developed by the Swedish Meteorological and Hydrological Institute, the Royal Netherlands Meteorological Institute, and the Danish Meteorological institute), simulated over three distinct representative concentration pathways: the RCP 2.6, RCP 4.5 and RCP 8.5 (Field, 2014). We use the resulting nine precipitation and temperature trajectories to force an HBV model (Lindström et al., 1997) validated over the control period, generating nine inflow trajectories. We H is the project horizon. As a result, K additional inflow from Corumana scenarios are generated. In each scenario, the inflow from Corumana on a day t is computed as: 4. Urban water demand uncertainty growth for the city of Maputo to be up to 2% per year. Urban demand multipliers are therefore assumed here to be constant every year, and a sample of size K is extracted from an uniform distribution within the range [1, 1.02]. Therefore, the urban demand trajectory in a certain scenario can be computed as the exponential increment of the historical urban demand. (cardinality of Ξ).

Robustness metric 265
To select the most robust alternative for each of the stakeholders, we used the minimax robustness metric. The computation of the metric requires N u simulations, one for each SOW χ ∈ Ξ.
The minimax identifies, among the optimal control policies π * designed via EMPODPS, the most robust alternative π r ∈ π * as the one attaining the best performance in the worst among the SOW:

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This metric, usually associated with a high risk aversion attitude, selects the alternative assuming that the worst future conditions will be realized (Wald, 1950).

Sensitivity and Uncertainty
In many hydrological applications, sensitivity and uncertainty analysis are often closely related. Yet, while the latter is used for quantifying the uncertainty in the output (i.e. the objective function), the former is typically adopted to apportioning output 275 uncertainty to the different uncertainty sources (or input factors) (Saltelli et al., 2008). According to , the two techniques often offer a valuable complement to each other, with SA providing valuable extra insights to UA on the identification of the most relevant uncertainty sources.

GLUE
We perform the quantification of the output variability in response to the four uncertainty sources considered in the paper by 280 the GLUE method. The implementation of GLUE encompasses several steps, most of which are already included within either the robustness or the sensitivity analysis.
A brief summary is provided below: 1. Generation of the States of the World: as described above, SOW are obtained in this study by near-random sampling of the perturbed time series of inflow, irrigation demand, infrastructure and population.

PAWN
To identify the main sources of vulnerability across robust policies, we use the PAWN sensitivity analysis (Pianosi and Wagener, 2015). PAWN is a distribution-based method, and its choice lies in its applicability to nonlinear models (Amaranto et al., 2020), and its independency from the type of output distributions (for example, symmetric, multimodal or highly skewed). In addition, 295 several studies (Zadeh et al., 2017;Pianosi and Wagener, 2018) have shown the capacity of PAWN to provide stable results for relatively low sample sizes.
In PAWN, the sensitivity of the output y (in this specific case, the objective function value) to variations of an input x i (climate, agriculture, infrastructure and population) is measured as the distance between the unconditional and conditional empirical cumulative distribution (CDF) of y. The distance between distributions is measured by the Kolmogorov-Smirnov 300 statistics, computed as follows: where F y (y) is the empirical unconditional distribution of y, and F (y|xi) (y) is the empirical conditional distribution of y when the i−th input is kept fixed at the nominal value x i . Considering the dependence of KS on the nominal value, the PAWN method considers KS statistics over a prescribed number of nomimal values and then computes the sensitivity index as follows:  Operatively, F y (y) is computed for a certain policy by iteratively simulating the system for N u SOW, generating therefore N u realizations of each objective upon which the empirical CDFs are constructed ( Figure 4). The computation of F (y|xi) (y) requires N c SOW simulations for each of the n c nominal value of x i (as shown in Figure 4 for x i = inf low). The unconditional empirical CDF for each uncertainty source is therefore estimated upon (n c · N c ) system realizations.
Considering M uncertainty sources, the total number of system evaluations required for implementing PAWN for a single 315 policy is therefore N u + (M · n c · N c ), where N u and N c are the number of SOW used for the unconditional and conditional distributions, respectively, while n c is the number of nominal value for the uncertainty sources.
In this study, we fixed N u , N c and n c to 5000, 2000 and 12. The decision is based on the observation of the uncertainty bounds in the sensitivity index obtained by 100 bootstraps of the input-output realizations upon which it is computed.

Optimization: Multiobjective Tradeoffs
The Pareto-optimal policies obtained by solving the optimization problem defined in equation 10 are reported in Figure 5.
We run the optimization on the baseline: i.e. the historical hydrological (∆ inflow = 0), agricultural (∆ irrigated area = 0), infrastructural (no external supply) and social (population growth = 0) conditions. Each line is a different policy, each axes represents an optimization objective, and the crossing point identifies the objective value derived from the implementation of 325 a certain policy (normalized between minimum and maximum), and to be minimized. The ideal solution is a horizontal line intersecting all four axes at their bottom. The extent of the conflicts is proportional to the slope of the lines connecting two adjacent axis. The colour of the lines represent the most robust policy for each stakeholder, and will be discussed more in details later.
Not surprisingly, the most conflicting objectives under the baseline scenarios are hydropower and upstream irrigation, since 330 the latter is the only stakeholder that cannot benefit of the dam releases from hydropower generation, and subtracts a potential source of release for energy production by pumping out of the reservoir to satisfy crop requirements. The combined urban and irrigation demand downstream is about 50% the turbine maximum capacity in the power plant, and can therefore be satisfied by hydropower releases. Compromise solutions between upstream irrigation, urban and downstream irrigation can be achieved at the expenses of 335 hydropower (as can be seen by the purple line in Figure 5), with a 10% reduction in energy production leading to a near-zero deficit for all the other stakeholders. The red line identifies a generic non robust (NR) policy, which generates the 90 th percentile in the overall robustness ranking.

Robustness: Probabilistic Tradeoffs
Even though historically NR led to objective values close to those of the most robust solution for upstream irrigation (RIU, green lines in Figures 5 and 6  overall the stakeholder less vulnerable to deep uncertain scenarios. This is also observable by the high density of grey policies close to the zero-deficit even in the worst scenario. In addition, about 80% of the optimal policies still ensured an irrigation deficit value within the historical deficit range (0, 102 m 3 s −1 Y −1 ) By observing the cloud of gray CDF's, one could notice how some of the operating policies do not monotonically increase towards higher objective values, but instead shows sudden discontinuities. Such discontinuities are probably due to the real-365 ization of states of the world causing the reservoir level to reach the lower limit of the discretionary operating space. In other words, the variation in the external forcings (most likely a decrease in inflow) is such that, for certain NR operating policies, the reservoir level triggers a zero-release condition for a certain time-period. The absence of releases is reflected in the objective value, which suddenly increases due to no supply availability to cover the water demand.
Finally, the objective function of RHP shown in Figure 6b4 rapidly increases even for more small system perturbations, making hydropower the most vulnerable sector to deep uncertainties. The possible explanation is twofold: the hydropower sector has the higher water demand, and is solely dependent on inflow. Both conditions makes it particularly sensitive to adverse hydroclimatic scenarios, while the latter further exacerbate its sector vulnerability due to the absence of favourable infrastructural measures to mitigate water scarcity impact.

375
To allow a comparative discussion on how different SOW realizations unevenly shape the vulnerability of different control policies, we select the most and the least robust among the operational alternatives described above for each sector, and analyze the corresponding sensitivities and uncertainties. As mentioned before, hydropower solely depend on inflow, and qualitative exhibits the same behavior for all the operating policies. For this reason, results for this sector are not further discussed here.
Figures 7a and 7b represent a scatterplot of the irrigation objective in correspondence with its relative forcing perturbation 380 for RID (most robust alternative) and RUD (least robust alternative), respectively. Colours identify, for a specific policy, those SOW whose realization causes the objective function to assume values below the fifth percentile (defined as the threshold values for the behavioral perturbation set). The robustness of RID against uncertain future condition is highlighted by the zero deficit produced in more than four thousands out of five thousands input realizations, and by the irrigation deficit below 1 m 3 s −1 Y −1 in the remaining (Figure 7a). Furthermore, inflow represents the only influential factor for this control policy (as 385 can be seen in Figure 7a1). In particular, irrigation deficit is produced only when the streamflow multiplier falls below 0.65.
This practically means that, for any future climatic conditions generating streamflow reduction below to 35%, RID is able to ensure water supply no matter the expansion in irrigated area. RID and RUD stands in how an expansion in irrigated area might be perceived: as an opportunity in the first case, and as a potential source of conflicts in the latter.
As far as the urban deficit is concerned, the increase in the objective value during the 2010 drought (Figures 8a3 and 8b3) highlights the pivotal role of the Greater Maputo Water Supply Project for ensuring continuity of supply. However, when RUD is adopted, urban deficit stabilizes around 80 m 3 s −1 Y −1 immediately after the event, and remains constant even if the project 400 is not completed by the end of the simulation horizon. In other words, if the infrastructure is built before the drought, little or no deficit is generated. Otherwise, the system recovers from the event, and afterwards urban water demand is fulfilled no matter and NR suggests that the structural intervention in the former case is a fundamental action to undertake in order to cope with extreme hydrological scenarios, while in the latter it becomes pillar for the everyday operation of the system, especially when population growth increases the water demand in the metropolitan area of Greater Maputo. Finally, upstream irrigation experiences a sudden increase in irrigation deficit as soon as inflow decreases and irrigation demand grows (Figures 9a and 9b). Unlike downstream irrigation districts, the upstream agricultural sector, not being able to 're-use' water from other sectors, is less flexible to a change in the external drivers, and therefore less robust to their uncertain realizations. However, when comparing RUI ( Figure 9a) and RHP (Figure 9b), it is evident that irrigation deficit grows unevenly among policies: it is dramatically high (over 500 m 3 s −1 Y −1 ) in the latter, and substantially lower in the former.
The higher sensitivity (SI > 0.9) of upstream irrigation deficit to agricultural expansion occurring when RHP is adopted (Figure 9d) seems to identify a systematic pattern in the comparison among most and least robust policies. For all the stakeholders 415 analyzed, the latter have not just been consistently vulnerable to inflow decrease, but also to the other non-hydroclimatic system perturbations. In other words: (1) infrastructural interventions become a must, as reflected by the increased urban deficit when the GMWSEP is not built (Figure 8b3); (2) population growth exerts a non-negligible pressure on the water system, and contributes towards increasing urban deficit ( Figure 8d); and (3) agricultural expansion is consistently limited by lack of water availability (Figures 9b and 9d). The opposite is true for RP: in spite of being (as expected) vulnerable to a decrease in inflow 420 (even if in lower magnitude with respect to NR policies), they rely on new infrastructure only in emergency conditions, and are able to sustain increases both in terms of population and irrigated area.

Conclusions
In this study, we implement an integrated decision-analytic framework combining optimization, robustness, sensitivity and uncertainty analysis to better understand the major sources of uncertainty for water supply strategies in the lower Umbeluzi 425 river, Mozambique. Results provide important insights on the robustness and vulnerability of reservoir operation to exogenous perturbations in managing multiple, conflicting objectives.
In particular, the main findings of this paper are: -Optimal reservoir operating policies exploring similar tradeoffs in current conditions might lead to substantially different results under deeply uncertain scenarios. Specifically, the non-robust optimal solution presented in this study was largely 430 dominated by 90% of the operating policies across all objectives once the system is perturbed.
-In water scarcity conditions, some new (i.e. non detected under historical conditions) tradeoffs between downstream irrigation and the urban supply suddenly emerge. In fact, while it is possible to explore operating policies ensuring maximum satisfaction for both stakeholders in current conditions, the two stakeholders are found in systematic competition, with the most robust policy for Greater Maputo being in the 95th percentile (96th out of hundred) when ranked for 435 downstream irrigation.
-Overall, downstream irrigation appeared to be the least vulnerable stakeholder, with about 60% of control policies ensuring an objective value in the worst possible condition lower than the maximum computed by forcing the system with the historical trajectories.
-Robust policy for downstream irrigation ensures sustaining agricultural production with near-zero deficit for the majority be a streamflow reduction, which however produces only a marginal deficit increase even in the worst condition. The opposite holds true when choosing the less robust policy, with objective values suddenly increasing even for small perturbations in streamflow and irrigation demand.
-As expected, hydropower production resulted to be solely dependent on inflow realizations. Furthermore, it was the least 445 robust sector among those considered, with fast decrease in objective function values as soon as the system is perturbed.
-The implementation of the greater Maputo water supply expansion project appears to be vital for sustaining urban water supply. However, while its role can be envisioned solely as 'drought mitigator' when adopting robust policies (even for maximum population growth rates), it becomes essential to mitigate day to day deficits for non-robust solutions.
-Overall, it is possible to conclude that robust policies are usually vulnerable only to hydrological perturbations and 450 are able to sustain the majority of population growth and agricultural expansion scenarios. Moreover, infrastructural interventions becomes crucial only in extreme drought conditions. On the contrary, non-robust policies are sensitive also to social and agricultural changes, and require structural interventions to ensure stable supply.
One of the main limitation of this study is the assumption of independence among the distributions used for the generation of the historical trajectories multipliers which constitute the foundation for developing the states of the world. Even though such 455 assumption allows to explore the full range of variability of the exogenous drivers of the system, and has been successfully applied in recent hydrological applications (see for example Amaranto et al. 2020), a study entangling the covariance among the uncertainty sources could provide further insights on the robustness of each operating policy, and can tailor the sensitivity analysis on a more reliable set of perturbations. It is therefore a recommendation for a future research exercise. Furthermore, robust policies are selected here according to the minimax robustness metric. minimax,