Assessing the robustness of a water resource system's performance under climate change involves exploring a wide range of streamflow conditions. This is often achieved through rainfall–runoff models, but these are commonly validated under historical conditions with no guarantee that calibrated parameters would still be valid in a different climate. In this note, we introduce a new method for the statistical generation of plausible streamflow futures. It flexibly combines changes in average flows with changes in the frequency and magnitude of high and low flows. It relies on a three-parameter analytical representation of the flow duration curve (FDC) that has been proved to perform well across a range of basins in different climates. We rigorously prove that, for common sets of streamflow statistics mirroring average behaviour, variability, and low flows, the parameterisation of the FDC under this representation is unique. We also show that conditions applied to these statistics for a solution to exist are commonly met in practice. These analytical results imply that streamflow futures can be explored by sampling wide ranges of three key flow statistics and by deriving the corresponding FDC in relation to model basin response across the full spectrum of flow conditions. We illustrate this method by exploring in which hydro-climatic futures a proposed run-of-river hydropower plant in eastern Turkey is financially viable. Results show that, contrary to approaches that modify streamflow statistics using multipliers applied uniformly throughout a time series, our approach seamlessly represents a large range of futures with increased frequencies of both high and low flows. This matches expected impacts of climate change in the region and supports analyses of the financial robustness of the proposed infrastructure to climate change. We conclude by highlighting how refinements to the approach could further support rigorous explorations of hydro-climatic futures without the help of rainfall–runoff models.

Projections of climate change and its impact on water resources are inherently uncertain, and this is likely to increase as a result of climatic, technological, economic, and sociopolitical changes

First, a predict-then-act approach is not compatible with hard-to-quantify uncertainties as it works best when a known single-probability density function is available for each key parameter

The second category of issues is with the use of rainfall–runoff models to generate future flow conditions. Indeed, these models have generally been calibrated and validated under historical conditions, with no assurance that these parameters would still be valid under different hydro-climatic conditions

For these reasons, approaches aimed at finding climate-robust adaptation solutions have often relied on multipliers applied uniformly along a time series also known as the delta change approach

As the representation of the empirical cumulative distribution function (CDF) of streamflow

This remark has led

This paper leverages the existence of high-performing parameterisations of the FDC across a range of climates to statistically generate plausible streamflow futures.
We directly link parameter triplets of the Kosugi model with the following three streamflow statistics that are relevant to the management of water resources: central tendency, variability, and low-flow indicator. This one-on-one correspondence enables us to (1) sample hydro-climatic futures according to plausible ranges for streamflow statistics and (2) convert these into ensembles of FDCs that represent
the differentiated impacts of climate change across flow quantiles. The latter is consistent with studies of historically observed streamflow change

This section demonstrates the technique that is the core of this paper and introduces its workflow. First, Sect.

The flow duration curve (FDC) is a cumulative-frequency curve that ranks the observed record of

To fit the Kosugi model and capture flow variability within the FDC, it is necessary to have daily discharge measurements over a sufficient period of time, e.g. more than 20 years.

In this paragraph, we directly relate the three parameters of the Kosugi FDC model with sets of three streamflow statistics that are of interest to water resource management. This is key to relating a hydro-climatic future (described with different flow statistics) to a well-defined FDC.
The central tendency and the spread or the degree of variation are the two key aspects of describing a distribution

We can entirely define the flow distribution associated with a hydro-climatic future defined by

Step-by-step derivation of these equations, along with proof of the uniqueness of a parameterisation, and conditions for the existence of solutions are provided in the Supplement to this paper. In this section, we provide the main results for both the mean and median cases.

In the mean case, we know

In the median case, we know

In this paragraph, we explain what the conditions for the existence and uniqueness provided imply – see Eqs. (

From Eq. (

Figure

Flowchart of the approach: (1) Kosugi model parameters are calibrated with a historical FDC; (2) a set of scenarios with modified flow statistics are determined; (3) a new set of Kosugi model coefficients are derived for each future scenario, and future scenarios are created using these coefficients; (4) future scenarios can be used in robustness assessments.

To generate future flows, one needs to sample a set of futures in step (2). This corresponds to sampling the chosen parameters

Note that the first three steps of this workflow can be replicated for any site using the Zenodo repository

This section demonstrates the fitness of our method for robustness assessments.

The case study involves the climate change impact analysis of a proposed run-of-river (RoR) hydropower plant at the Besik site on the Mukus River in the province of Van located in the eastern Anatolia region of Turkey (38.15

Contrary to reservoir-based hydropower plants, RoR schemes have virtually no storage, so they are vulnerable to changes in flow as they cannot modulate flows and only operate in a predefined range. Extreme low flows are insufficient to activate the turbines, and equally, flows above the design discharge do not produce additional energy. Because of this focus on the mid-range flows, the median is a more important indicator of performance than the mean flow, which can be skewed by high discharges. For this reason, this application will relate the median, coefficient of variation, and first-percentile flows to Kosugi parameters (the median case).

In step (1) of our approach, we fit the three-parameter Kosugi model to the daily discharge data.
Figure

Plot of the daily flow duration curves (FDCs) used in the case study (red circles). Black line represents the fitted Kosugi model, and the blue line is the FDC deduced from

In step (2), we determine plausible ranges for the three statistical parameters over the operational life of the proposed plant. In Turkey, hydropower projects are licensed to generate electricity for a period of 49 years.
Several climate projections indicate a decrease in the mean discharge values that could reach up to 60 %

Sampling ranges for multipliers of statistical parameters, where 1 corresponds to the values for the historical time series.

Next, in step (3), we primarily check if the samples satisfy the condition for existence; the smallest and largest measured values of

Plot of the flow duration curves (FDCs) of the historical record (blue line) and sampled flow duration curves (grey lines) constructed by deriving the FDC parameters for the Kosugi model shown in Table

Finally, in step (4), we evaluate the performance of a design under generated future flows.
We input each ensemble member into state-of-the-art software to compute the technical performance, energy production, and economic profit of a design with a given site's characteristics

Plot of generated flow duration curves (FDCs), with each solution coloured by its net present value (NPV). Grey-coloured lines signify states of the world (SOWs) in which NPV is negative. NPV of the optimal design based on observed discharge (blue line) is USD 10 M.

In this technical note, we present an effective, practical, and novel approach based on a near-universal parameterisation of flow duration curves (FDCs) and perturbation of these parameters to simulate a range of futures in a way that is hydrologically consistent across the spectrum of hydrological conditions. Our application to a run-of-river hydropower project in eastern Turkey showcases the ability of our method to provide a large range of climate-modified catchment responses, including increased frequency of both high flows and low flows to mimic the future projections for the area (i.e. more arid conditions with increased trends of extreme hydrological events). It compares favourably with existing statistical methods for perturbing flows, such as the delta change approach. This then supports robustness analyses for rivers for which no detailed hydrological model is available (applied here to assess the financial viability of run-of-river hydropower design in a changing climate). The ease of application of the method illustrates its wide applicability in terms of supporting robustness assessments of infrastructure where streamflow variability impacts performance. We now conclude with some remarks on how this novel approach could be extended to further support such assessments.

Even though the three-parameter Kosugi model has been shown to fit FDCs well across a wide range of catchment characteristics

Our method focuses on catchments free of major flow regulation (reservoir, effluent discharge). Yet, those catchments do not have to be pristine and can, for example, experience significant human interference in land use change. Indeed, the MOPEX dataset

We also identified two current limitations to this method that we believe can be addressed by future developments.
First, recent studies reveal that there is an increasing trend in the number of zero-flow days in many regions such as the Mediterranean

Our approach only considers the FDC and says nothing of the seasonality, frequency, and duration of dry and wet spells. The shifting seasonality of flows in a changing climate can easily be captured by combining our approach with methods such as the log space rescaling of stationary flows

The climate-perturbed FDC generation model has been developed in Python 3.10.4 and is provided with an environment file. It is accessible from the Zenodo open-access repository at

The supplement related to this article is available online at:

VY: conceptualisation, methodology, writing – original draft, formal analysis, software development, investigation. CR: supervision, conceptualisation, methodology, formal analysis, writing – review and editing, investigation. RM: investigation. SB: conceptualisation, investigation, supervision. All the authors have read and agreed to the published version of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Solomon Brown and Robert Milton were supported by the UK Engineering and Physical Sciences Research Council (EPSRC) through the “Table Top Manufacturing of Tailored Silica for Personalised Medicine [SiPM]” project (reference no. EP/V051458/1). We also appreciate the insights and comments from the associate editor Micha Werner and from two anonymous referees as they have greatly improved this paper. The first author gratefully acknowledges support from the General Directorate of State Hydraulic Works (DSI-TURKEY).

This research has been supported by the UK Engineering and Physical Sciences Research Council (grant no. EP/L016818/1).

This paper was edited by Micha Werner and reviewed by two anonymous referees.