Developing a Robust Multi-Attribute Decision-Making Framework to Evaluate Performance of Water System Design and Planning under Climate Change

Abstract

In theory, emergence of robustness concept has pushed decision-makers toward designing alternatives, such as resistant against the potential fluctuations fueled by uncertain surrounding environment. This study promotes an objective-based multi-attributes decision-making framework that takes into account the uncertainties associated with the impacts of the climate change on water resources systems. To capture the uncertainties of climate change, Monte Carlo approach has been used to generate a series of ensembles. These generated ensembles represent the stochastic behavior of the hydro-climatic variables under climate change. This framework represents the inherent uncertainties associated with hydro-climatic simulations. Next, a coupled TOPSIS/Entropy multi-attribute decision-making framework has been formed to prioritize the feasible alternatives using system performance measures. The main objective of this framework is to minimize the risk of deceptive and subjective assessments during decision-making process. Karkheh River basin has been selected as a case study to demonstrate the implication of this framework. Using a set of system performance attributes, the performance of two hydropower systems has been estimated during the baseline period and under the future climate change conditions. According to the conducted frequency analysis, the alternative in which both hydropower projects would go under construction emerged as the robust solution (i.e., there was a 99.9% chance that it outperforms other solutions). The results indicate that the construction of these hydropower systems leads to the increase of Karkheh River basin robustness in the future.

Appendix

1.1 Downscaling Method: Change Factor Formulation

The change factor technique is utilized for simple spatial downscaling. In this simple weather typing approach, the climate scenarios are obtained by computing the differences (or ratio, depending on the nature of the climate variables) between the averages of the GCM dataset for the climate change period and the corresponding averages of the model’s simulation results for the baseline period. The changes in air temperature are usually expressed as differences, whereas for rainfall changes, ratios are commonly used. The air temperature and rainfall changes in the change factor downscaling procedure can be expressed as:

$$ \varDelta {T}_i={\overline{T}}_i^{fut}-{\overline{T}}_i^{base} $$

(A1)

$$ \varDelta {P}_i=\frac{{\overline{P}}_i^{fut}}{{\overline{P}}_i^{base}} $$

(A2)

$$ {T}_i={T}_i^{obs}+\varDelta {T}_i $$

(A3)

$$ {P}_i={P}_i^{obs}\times \varDelta {P}_i $$

(A4)

in which ΔT_i and ΔP_i = average long-term monthly air temperature and rainfall changes for month i, respectively; ${\overline{T}}_i^{fut}$ and ${\overline{P}}_i^{fut}$ = average long-term monthly air temperature and rainfall for month i simulated by the GCM for the climate change period, respectively; ${\overline{T}}_i^{base}$ and ${\overline{P}}_i^{base}$ = average long-term monthly air temperature and rainfall for month i simulated by the GCM for the baseline period, respectively; ${T}_i^{obs}$ and ${P}_i^{obs}$ = observed air temperature and rainfall for month i, respectively; and T_i and P_i = air temperature and rainfall for month i in the climate change period, respectively.

1.2 Hydropower Simulation

Hydropower simulation, naturally, is the prerequisite step to quantify the impacts of climate change on a hydropower system. Thus, modeling of a hydropower system includes simulations of both water and power, which can be expressed as follows:

$$ {S}_{\left(r,t+1\right)}={S}_{\left(r,t\right)}+{Q}_{\left(r,t\right)}+{M}_{n\times n}R{e}_{\left(r,t\right)}-S{p}_{\left(r,t\right)}- Los{s}_{\left(r,t\right)}\kern0.5em r=1,\dots, n\operatorname{Re}s;t=1,\dots, N $$

(A5)

$$ Los{s}_{\left(r,t\right)}={A}_{\left(r,t\right)}\times E{v}_{\left(r,t\right)} $$

(A6)

$$ {A}_{\left(r,t\right)}=G\left[{S}_{\left(r,t\right)}\right] $$

(A7)

$$ S{p}_{\left(r,t\right)}=\frac{\left|S{\max}_r-A{W}_{\left(r,t\right)}\right|-\left[S{\max}_r-A{W}_{\left(r,t\right)}\right]}{2} $$

(A8)

$$ A{W}_{\left(r,t\right)}={S}_{\left(r,t\right)}+{Q}_{\left(r,t\right)}+{M}_{nRes\times nRes}{\operatorname{Re}}_{\left(r,t\right)}- Los{s}_{\left(r,t\right)} $$

(A9)

$$ 0\le \operatorname{Re}\left(r,t\right)\le R{\max}_r $$

(A10)

$$ Smi{n}_r\le {S}_{\left(r,t\right)}\le S{\max}_r $$

(A11)

$$ P{P}_{\left(r,t\right)}=\frac{\gamma_W\times g\times {\eta}_r\times \varDelta {H}_{\left(r,t\right)}\times R{e}_{\left(r,t\right)}}{86^{\prime }400\times Countda{y}_t\times {n}_r} $$

(A12)

$$ \varDelta {H}_{\left(r,t\right)}={H}_{\left(r,t\right)}-T{W}_{\left(r,t\right)} $$

(A13)

$$ {H}_{\left(r,t\right)}=F\left[{S}_{\left(r,t\right)}\right] $$

(A14)

$$ T{W}_{\left(r,t\right)}=Y\left[R{e}_{\left(r,t\right)}\right] $$

(A15)

in which, S_(r,t) and S_{(r,t + 1)} = stored water volume in the rth reservoir at the tth and (t + 1)th time-step, respectively; Q_(r,t) = upstream inflow volume to the rth reservoir at the tth time step; M_{nRes × nRes} = reservoirs’ connection matrix; Re_(r,t) = released water volume from the rth reservoir at the tth time step; Sp_(r,t) = spilled water volume from the rth reservoir at the tth time step; Loss_(r,t) = lost volume of water due to evaporation for the rth reservoir at the tth time step; A_(r,t) = water surface area of the rth reservoir at the tth time step; Ev_(r,t) = evaporation water depth of the rth reservoir at the tth time step; G[] = simulating function of the stored water’s surface area; AW_(r,t) = available water in the rth reservoir at the tth time step; Smax_r = storage capacity of the rth reservoir; Rmax_r = maximum water volume released from the rth reservoir within a time step; Smin_r = dead storage of the rth reservoir; PP_(r,t) = generated hydropower by the rth reservoir at the tth time step; γ_W = water specific weight; g = gravitational acceleration; η_r = efficacy of the rth reservoir’s hydropower system; ΔH_(r,t) = height difference between the reservoir water level and the tailwater level of the rth reservoir at the tth time step; Countday_t = number of days within the tth time step; n_r = plant factor for the rth reservoir’s hydropower system; H_(r,t) = water level of the rth reservoir at the tth time step; TW_(r,t) = height of downstream water for the rth reservoir at the tth time step; F[] = simulating function of the water level; Y[] = simulating function of the tailwater level; and nRes = total number of reservoirs.

1.3 Shannon’s Entropy

Shannon’s Entropy is an objective-based assignment mechanism, and the mathematical structure to implement Shannon’s Entropy is as follows:

$$ {p}_{ij}=\frac{x_{ij}}{\sum \limits_{i=1}^m{x}_{ij}}\kern0.5em j=1,\dots, n $$

(A16)

$$ {E}_j=-\frac{1}{\ln {m}_j}\sum \limits_{i=1}^m{p}_{ij}\times \ln {p}_{ij} $$

(A17)

$$ {d}_j=1-{E}_j $$

(A18)

$$ {w}_j=\frac{d_j}{\sum \limits_{j=1}^n{d}_j} $$

(A19)

in which, x_i,j = the rank of the ith class with regard to the jth evaluation criterion; E_j = The entropy of the jth criterion; m = the number of classes for each given criterion; d = the distance of a given criterion from the entropy; and w = the assigned weight to each criterion.

1.4 TOPSIS

The basic computation algorithm of TOPSIS is as follows:

Step I | Constructing the original decision matrix.

Step II | Defining the reference alternatives (i.e., the ideal solution and the negative-ideal solution). The ideal alternative is an arbitrarily defined vector, which describes the aspired solution to the given problem, while the inferior alternative is an arbitrarily defined solution that represents the most undesirable option for the given MADM problem.

Step III | Normalizing the decision-matrix.

Step IV | Creating the weighted normalized decision-matrix. The assigned weights to the evolution criteria reflect their relative importance to the decision-makers and can be obtained by objective approaches such as Shannon’s Entropy method. The higher these weights, the more crucial their roles would be in the selection process.

Step V | Computing the distance of every given alternative to the reference points, namely referred to in TOPSIS as the separation measurement. These distances would help decision-makers to quantify the relative closeness of each alternative to the ideal solution, which can be used to rank the desirability of the feasible alternatives.

1.5 Weighted Vulnerability

The weighted vulnerability is given by Eq. (9):

$$ {\upsilon}_W=\sum \limits_{t\in F}{Se}_t\times {e}_t $$

(A20)

in which, e_t = the weight of a failure event at time t. To do so, a nonparametric distribution is introduced in Eq. (A10) to replace e_t in Eq. (9), yielding the weighted vulnerability written in Eq. (A11):

$$ {e}_t=\frac{Se_t}{\sum \limits_{t\in F}{Se}_t}\kern0.5em t=1,2,\dots, T $$

(A21)

$$ {\upsilon}_W=\sum \limits_{t\in F}\frac{{\left({Se}_t\right)}^2}{\sum \limits_{t\in F}{Se}_t} $$

(A22)

in which, Se_t = severity of the failure in period t.

1.6 Probability Distribution Functions

• Normal Distribution:

The pdf of the normal distribution for the variable x is:

$$ f\left(x|\mu, \sigma \right)=\frac{1}{\sqrt{2{\pi \sigma}^2}}{e}^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma}^2}} $$

(A23)

in which, μ = the mean (i.e., the location parameter); and σ = the standard devition (i.e., scale parameter).

• Log-normal & 3-parameter log-normal Distributions:

The pdf of the log-normal distribution is:

$$ f\left(x|\mu, \sigma \right)=\frac{1}{\sqrt{2\pi {x}^2{\sigma}^2}}{e}^{-\frac{{\left(\ln x-\mu \right)}^2}{2{\sigma}^2}} $$

(A24)

The 3-parametr version of this distribution is:

$$ f\left(x|\mu, \sigma, \theta \right)=\frac{1}{\sqrt{2\pi {\left(x-\theta \right)}^2{\sigma}^2}}{e}^{-\frac{{\left[\ln \left(x-\theta \right)-\mu \right]}^2}{2{\sigma}^2}} $$

(A25)

in which, θ = the threshold parameter.

• Exponential Distribution:

The pdf of the exponential distribution is:

$$ f\left(x|\lambda \right)=\left\{\begin{array}{cc}\lambda {e}^{-\lambda x}& x\ge 0\\ {}0& x<0\end{array}\right. $$

(A26)

in which, λ = the rate parameetr (i.e., scale parameter).

• Weibull & 3-parameter Weibull Distributions:

The pdf of the Weibull distribution is:

$$ f\left(x|\lambda, \xi \right)=\left\{\begin{array}{cc}\frac{\xi }{\lambda }{\left(\frac{x}{\lambda}\right)}^{\xi -1}{e}^{-{\left(\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.\right)}^{\xi }}& x\ge 0\\ {}0& x<0\end{array}\right. $$

(A27)

in which, ξ = the shape parameetr. The 3-parametr version of this distribution is:

$$ f\left(x|\lambda, \xi, \theta \right)=\left\{\begin{array}{cc}\frac{\xi }{\lambda }{\left(\frac{x-\theta }{\lambda}\right)}^{\xi -1}{e}^{-{\left(\raisebox{1ex}{$x-\theta $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.\right)}^{\xi }}& x\ge \theta \\ {}0& x<\theta \end{array}\right. $$

(A28)

• Extreme Value Distribution:

The pdf of the extreme value distribution is:

$$ f\left(x|\mu, \sigma, \xi \right)=\left\{\begin{array}{cc}{\left[1+\xi \left(\frac{x-\mu }{\sigma}\right)\right]}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\xi $}\right.}& \xi \ne 0\\ {}0& x=0\end{array}\right. $$

(A29)

• Gamma & 3-parameter Gamma Distributions:

The pdf of the gamma distribution is:

$$ f\left(x|\xi, \lambda \right)=\frac{1}{\left(\xi -1\right)!{\lambda}^{\xi }}{x}^{\xi -1}{e}^{-\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.} $$

(A30)

The 3-parametr version of this distribution is:

$$ f\left(x|\xi, \lambda, \theta \right)=\frac{1}{\left(\xi -1\right)!{\lambda}^{\xi }}{x}^{\xi -1}{e}^{-\frac{\left(x-\theta \right)}{\lambda }} $$

(A31)

• Logistic Distribution:

The pdf of the logostic distribution is:

$$ f\left(x|\mu, \sigma \right)=\frac{e^{-\frac{x-\mu }{\sigma }}}{\sigma {\left(1+{e}^{-\frac{x-\mu }{\sigma }}\right)}^2} $$

(A32)

• Log-logistic Distribution:

The pdf of the log-logostic distribution is:

$$ f\left(x|\lambda, \xi \right)=\frac{\left(\frac{\xi }{\lambda}\right){\left(\frac{x}{\lambda}\right)}^{\xi -1}}{{\left[1+{\left(\frac{x}{\lambda}\right)}^{\xi}\right]}^2} $$

(A33)