# Optimizing Water Allocation under Uncertain System Conditions for Water and Agriculture Future Scenarios in Alfeios River Basin (Greece)—Part B: Fuzzy-Boundary Intervals Combined with Multi-Stage Stochastic Programming Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}of water allocated to each one of the water uses has been undertaken for the Alfeios River in [38].

## 2. Mathematical Formulation of the FBISP Method

#### Limitations of the Applied Methodology and Corresponding Changes

_{1}, s

_{2}, …, s

_{n}with respective probability masses p

_{1}, p

_{2}, …, p

_{n}with p

_{i}> 0 and $\sum _{i=1}^{n}{p}_{i}=1$. The expected value of the second-stage optimization problem can be written as the summation of the products of the values of each scenario with its probability mass. Based on this transformation, the TSP problem is expressed as a large linear programming problem forming the deterministic equivalent of the original problem. This approach has been further advanced and various techniques have been suggested in order to enable its efficient numerical solution [51,52].

^{K}. The number of realizations/scenarios of the random variables (or in case of continuous distribution the number of discretization points) typically grows exponentially with the dimensionality of the variables and therefore, this number can quickly become prohibiting for the computational capacities of modern computers. As analyzed in [53], a common technique for reducing the number of scenario set to a manageable size is by using Monte Carlo simulation through generation of a sample x

_{1}, x

_{2}, .., x

_{N}of replications N of the random variable. Given a sample x

_{1}, x

_{2}, .., x

_{N}of replications N, the expectation function is approximated by the sample average. By the Law of Large Numbers this average value converges pointwise to the expected value as N→∞. This approach is called Sample Average Approximation (SAA) method. The SAA problem is a function of the considered sample and in that sense is random. For a given sample x

_{1}, x

_{2}, .., x

_{N}, the SAA problem is of the same form as a two-stage stochastic linear programming problem with the scenarios s

_{1}, s

_{2}, …, s

_{n}each taken with the same probability equal to 1/N.

_{j}= a

_{−j}). This model reproduces the long-term persistence, and has been further generalized for application to simultaneous generation of stochastically dependent multiple variables. This is achieved by generating correlated (multivariable) white noise. At the second stage, the monthly synthetic values are generated posing emphasis on the reproduction of periodicity. A periodic first-order autoregression, abbreviated as PAR(1), model is used, which has been also generalized for multi-variable simulation. The final step is the coupling of the two time scales through a linear disaggregation model [55]. A brief description of the process for the generation of fifty short-time equal-probability scenarios simultaneously for the monthly rain and temperature variables and the corresponding hydrologic simulation for the computation of the discharges at the main four subcatchments of the Alfeios river basin is included in Section 3.2 and the detailed description is given in [38].

^{11}scenarios).

## 3. Formulation of Optimization Problem for the Alfeios River Basin

#### 3.1. Description of the Alfeios River Basin

^{3}/s) in the Peloponnese region in Greece. The main river and its six tributaries represent a significant source of water supply for the region, aiming at delivering and satisfying the expected demands from a variety of water users, including irrigation, drinking water supply, hydropower production and to a smaller extend recreation. A plethora of environmental stresses were exerted on its river basin the last decades. It drains an area of 3658 km

^{2}. Its annual water yield is estimated to be 2100 × 10

^{6}m

^{3}.

^{2}. The irrigation water demand extends officially from May to September, whereas in most years could be further extended from April up to October due to dry climatic conditions. The present monthly irrigation water demand is composed of two parts: (a) a regulated and stable irrigation demand pattern, referring only to the required water volume releases from Ladhon Reservoir, which is derived from the official agreement between Hellenic Public Power Corporation and General Irrigation Organization for the Flokas Irrigation Area; and (b) an extra uncertain irrigation demand at Flokas Dam site based on the actual crop patterns and the water inflows at this position. The total irrigation requirements for the crop pattern of Flokas are estimated for each stochastic hydrologic scenario using the FAO software CROPWAT 8.0. For the application of the FBISP methodology the upper- and lower-bounds of the water allocation targets for irrigation in EUR/m

^{3}are required. The optimized water allocation target for irrigation (Table 2) is explored, assuming that the irrigation demand can vary between the maximum demand of the present crop pattern and the maximum demand given in the study of the small HPS at Flokas. Based on this assumption, the lower-bound of the optimized water allocation target is set equal to the maximum of all sets of irrigation water requirements for the fifty hydrologic scenarios as computed by CROPWAT for the present irrigated area and crop pattern.

^{3}/s, the entire part of the river flow passes through the Flokas HPS, maintaining the water level of Dam at a stable level. When river flow rate exceeds 90 m

^{3}/s then the surplus flows over the spillways of Flokas Dam. Whereas for flood water volumes exceeding 300 m

^{3}/s the HPS Flokas closes for security reasons and the flood volume passes through the spillways of the dam and the opened gateways. For the application of the FBISP method the upper- and lower-bounds of the optimized hydropower production target ${T}^{\pm}$ (in MWh) at Flokas small HPS (Table 3) are required. These bounds are approximated also in this case, from the statistical analysis of the monthly timeseries of hydropower production at Flokas from 2011 to 2015. The ranges between the mean value of the historical timeseries minus its standard deviation (lower-bound) and its mean value plus its standard deviation (upper-bound) are taken into account as specified in [38].

^{3}/s for the drinking water supply system for the north and central part of the Region of Hleias is diverted from Erymanthos to the water treatment plant and then to the neighboring communities extending up to the city of Pyrgos. Due to the short operation period (starting in 2013), this water use is not incorporated in the optimization process as a variable but as a steady and known water abstraction demand.

**Table 1.**Upper- (THydroLadhon

^{+}) and lower- (THydroLadhon

^{−}) bounds of optimized target for hydropower production at HPS at Ladhon.

Bounds of Optimized Target | Target Limits for Hydropower Production at Ladhon HPS (MWh) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

January | February | March | April | May | June | July | August | September | October | November | December | Annual | |

THydroLadhon^{−} | 11,857 | 12,553 | 11,810 | 11,046 | 11,081 | 8965 | 9077 | 7613 | 5925 | 7387 | 9427 | 8540 | 115,282 |

THydroLadhon^{+} | 37,353 | 38,947 | 48,311 | 35,391 | 23,237 | 15,868 | 15,598 | 14,233 | 13,642 | 17,062 | 17,971 | 24,276 | 301,890 |

Time Stages | Irrigation Water Demand (m^{3}/s) | |
---|---|---|

Lower-Bound of Optimized Allocation Target | Upper-Bound of Optimized Allocation Target | |

Tirrigation^{−} | Tirrigation^{+} | |

t = 1—January | 0 | 0 |

t = 2—February | 0 | 0 |

t = 3—March | 0 | 6 |

t = 4—April | 2.0 | 6 |

t = 5—May | 5.0 | 6 |

t = 6—June | 8.9 | 12 |

t = 7—July | 11.5 | 12 |

t = 8—August | 9.2 | 12 |

t = 9—September | 2.7 | 6 |

t = 10—October | 1.2 | 6 |

t = 11—November | 0 | 0 |

t = 12—December | 0 | 0 |

Annual (m)^{3} | 108,756,934 | 174,700,800 |

**Table 3.**Upper- (THydroFlokas

^{+}) and lower- (THydroFlokas

^{−}) bounds of optimized target for hydropower production at HPS at Flokas.

Bounds of Optimized Target | Target Limits for Hydropower Production at Flokas HPS (MWh) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

January | February | March | April | May | June | July | August | September | October | November | December | Annual | |

THydroFlokas^{−} | 1244 | 1740 | 2450 | 2045 | 1574 | 437 | 219 | 218 | 232 | 395 | 299 | 1129 | 11,982 |

THydroFlokas^{+} | 2379 | 2894 | 3435 | 2840 | 1861 | 773 | 251 | 255 | 571 | 1111 | 1397 | 2097 | 19,865 |

#### 3.2. Optimization Problem of the Alfeios Hydrosystem

_{1}(${m}^{2}$);

_{1}for the water user i with i = 1, 2 corresponding to Ladhon and Flokas, respectively;

_{1}for the water user i with i = 1, 2 corresponding to Ladhon and Flokas, respectively;

^{+}model), and lower-bound (which are used in the f

^{−}model) are created. The steps of this process and the software programs used are presented schematically in the form of a flow chart in Figure 2. The last year of each of the fifty stochastic monthly discharge scenarios (since the future scenarios refer to ten years after the baseline) serves as input inflows into the optimization model for the optimal water allocation of Alfeios river basin. The monthly discharge at Flokas Dam, which is of interest for the optimization process, since at this position the available water is diverted to the irrigation canal, is computed as the sum of the four subcatchments as described in [38].

^{2}takes values ≥ 0.9.

^{3}), and secondly, the initial water level of Ladhon reservoir at stage zero (m

^{3}) which is expressed as deterministic-boundary interval (12,362,644.01, 26,783,729.12). A detailed analysis of the estimation of the prementioned unit benefits and penalties is provided in [38]. Briefly, based on the estimations of the Chief Engineer at Ladhon hydropower station, the upper- and lower- fuzzy-boundary intervals for the unit benefit and penalty of Ladhon are defined as given in Table 4. The shadow penalty for the hydropower production at Ladhon is composed of two parts: (a) the penalty due to hydropower shortage in comparison to the hydropower production target and (b) the penalty for the water spilled from the Ladhon Dam (if any) and is not available for hydropower production, which intends to express the opportunity loss of hydropower energy production. Based on the monthly selling price data of Flokas HPS, its unit benefit is approximated as a single deterministic interval (and not as a fuzzy-boundary interval) as presented in Table 4. The unit shadow penalty is approximated as a single deterministic interval and is taken equal to the upper-bound solution of the unit penalty of Flokas HPS, as shown in Table 4. The unit benefit for water allocated to irrigation is interpreted as the probable net income from agricultural production of the Flokas crop pattern, taking into account the brutto farmer, the cost of production, the cost of the irrigation canal (associated to the water charge to the farmers from the general irrigation organization) and the organizational structure of local irrigation organizations (the charges of the local irrigation organizations). In Table 5, the lower- and upper- fuzzy-boundaries of the unit benefit of water allocated to the Flokas irrigation scheme is provided for the baseline scenario and the WADI future scenarios. Finally, the unit penalty from the irrigation water deficits is based on the crop yield reduction and the corresponding net farmer income loss. In Table 6 the lower- and upper- fuzzy-boundaries of the unit penalties for irrigation water deficits of the Flokas irrigation scheme is provided for the baseline scenario and the WADI future scenarios (where as presented in Section 4 the World agricultural markets scenario is denoted as Future Scenario 1 (FS1), the Global agricultural sustainability scenario as Future Scenario 2 (FS2), the Provincial agriculture scenario as (Future Scenario 3-FS3) and the Local community agriculture as (Future Scenario 4-FS4).

**Table 4.**Lower- and upper- fuzzy-boundary intervals for the unit benefit and unit penalty for hydropower production EUR/MWh at Ladhon and at Flokas.

Variables | NBHP Ladhon | NBHP Flokas | CHP Ladhon | CHP Flokas |
---|---|---|---|---|

EUR/MWh | EUR/MWh | EUR/MWh | EUR/MWh | |

Lower-Bound—Minimum | 40 | 87.5 | 120 | 140 |

Lower-Bound—Maximum | 55 | – | 130 | 150 |

Upper-Bound—Minimum | 60 | 80 | 140 | 140 |

Upper-Bound—Maximum | 75 | – | 150 | 150 |

**Table 5.**Lower- and upper- fuzzy-boundary intervals for the unit benefit from irrigation for the baseline and the WADI future scenarios for Flokas irrigation scheme EUR/m

^{3}.

Fuzzy-Boundary Intervals | NBIrrigationFlokas EUR/m^{3} | |||||
---|---|---|---|---|---|---|

Baseline | FS 1 | FS 2 | FS 3 | FS 4 | ||

Upper-Bound | Min | 0.166 | 0.127 | 0.189 | 0.191 | 0.221 |

Max | 0.175 | 0.136 | 0.265 | 0.276 | 0.294 | |

Lower-Bound | Min | 0.187 | 0.190 | 0.266 | 0.277 | 0.295 |

Max | 0.205 | 0.234 | 0.269 | 0.314 | 0.431 |

**Table 6.**Lower- and upper- fuzzy-boundary intervals for the unit penalties for water deficits to irrigation EUR/m

^{3}for the baseline and the future scenarios.

Fuzzy-Boundary Intervals | PEIrrigationFlokas EUR/m^{3} | |||||
---|---|---|---|---|---|---|

Baseline | FS 1 | FS 2 | FS 3 | FS 4 | ||

Upper-Bound | Min | 0.989 | 0.748 | 1.052 | 1.035 | 1.043 |

Max | 1.051 | 1.159 | 1.075 | 1.073 | 1.070 | |

Lower-Bound | Min | 1.715 | 3.361 | 1.537 | 1.552 | 2.184 |

Max | 1.812 | 3.410 | 1.891 | 1.871 | 2.279 |

## 4. WADI Water and Agriculture Future Scenarios

Crop Prices | Baseline | World Agricultural Markets | Global Agricultural Sustainability | Provincial Agriculture | Local Community Agriculture | ||||
---|---|---|---|---|---|---|---|---|---|

Crops selling prices perceived by the farmers | – | Min | Max | Min | Max | Min | Max | Min | Max |

Maize | 100 | 85 | 95 | 95 | 105 | 100 | 110 | 100 | 110 |

Maize area subsidy | 100 | 0 | – | 75 | 85 | 90 | 100 | 85 | 95 |

Set aside quota | 100 | 0 | – | 95 | – | 100 | – | 105 | – |

Tomato | 100 | 85 | 95 | 110 | 120 | 100 | 110 | 120 | 130 |

Potato | 100 | 85 | 95 | 95 | 105 | 105 | 115 | 120 | 130 |

Watermelons | 100 | 85 | 95 | 95 | 105 | 105 | 115 | 120 | 130 |

Cotton | 100 | 80 | 90 | 90 | 100 | 85 | 95 | 110 | 120 |

Cotton subsidy | 100 | 0 | – | 85 | – | 90 | – | 105 | – |

Olive Trees | 100 | 80 | 90 | 85 | 95 | 90 | 100 | 100 | 110 |

Olive trees area subsidy | 100 | 0 | – | 95 | – | 95 | – | 105 | – |

Alfalfa | 100 | 80 | 90 | 90 | 100 | 100 | 110 | 110 | 120 |

Citrus | 100 | 85 | 95 | 95 | 105 | 100 | 110 | 120 | 130 |

Input prices | – | Min | Max | Min | Max | Min | Max | Min | Max |

Fertilizers | 100 | 85 | 100 | 140 | 150 | 100 | 110 | 150 | 160 |

Pesticides | 100 | 110 | 120 | 100 | 105 | 105 | 115 | 95 | 100 |

Energy | 100 | 85 | 95 | 120 | 130 | 100 | 110 | 130 | 140 |

Seeds | 100 | 100 | 110 | 110 | 120 | 120 | 130 | 130 | 140 |

Machinery | 100 | 100 | 115 | 115 | 135 | 100 | 115 | 120 | 140 |

Contractor services | 100 | 130 | 135 | 120 | 130 | 130 | 140 | 110 | 120 |

Water prices | 100 | 100 | 110 | 115 | 130 | 100 | 110 | 120 | 140 |

Irrigation infrastructure | 100 | 100 | 110 | 120 | 130 | 115 | 125 | 130 | 150 |

Labour | 100 | 90 | 100 | 100 | 110 | 95 | 105 | 110 | 120 |

Land | 100 | 110 | 120 | 110 | 125 | 100 | 110 | 85 | 95 |

Other inputs | 100 | 85 | 95 | 125 | 135 | 85 | 95 | 130 | 140 |

Crop yield changes due to technology | 100 | 110 | 120 | 100 | 115 | 100 | 105 | 85 | 105 |

Restriction on chemical use | 100 | 130 | 140 | 120 | 130 | 110 | 120 | 100 | 110 |

## 5. Results

^{α}composed of four options of maximized system benefits in combination with minimized probabilistic penalties corresponding to different system conditions. These four options for each solution method correspond to lower-min $\underset{\_}{{f}_{opt}^{-}}$ (and in tables and figures written as min (f

^{−})), lower-max $\underset{\_}{{f}_{opt}^{+}}$ (and in tables and figures written as max (f

^{−})), upper-min $\overline{{f}_{opt}^{-}}$ (and in tables and figures written as min (f

^{+})), upper-max $\overline{{f}_{opt}^{+}}$ (and in tables and figures written as max (f

^{+})). These results (four prementioned options), however, do not necessarily construct a set of stable intervals [2].

^{5}= 32 possible combinations of the uncertain variable values/runs of the algorithm for each WADI future scenario (32 × 5 = 160 runs for all examined scenarios in total) has been undertaken based on the FBISP algorithm as proposed by [2].

#### 5.1. Results Analysis for the Baseline Scenario

WADI Scenarios | Total Annual Net Benefit (EUR) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 96,192,950 | 102,180,847 | 127,801,604 | 135,950,230 | 104,523,859 | 109,324,450 | 128,786,579 | 134,978,247 |

FS1 | 76,667,389 | 81,865,504 | 113,991,044 | 126,002,601 | 84,858,261 | 89,043,374 | 114,966,693 | 125,209,412 |

FS2 | 127,853,717 | 147,770,816 | 179,855,724 | 187,230,205 | 140,082,873 | 157,717,364 | 181,108,510 | 185,941,332 |

FS3 | 100,450,661 | 119,991,812 | 143,479,720 | 154,964,308 | 108,428,316 | 126,072,379 | 144,472,397 | 154,016,575 |

FS4 | 139,941,260 | 159,235,851 | 194,409,861 | 225,532,763 | 152,798,364 | 169,543,642 | 195,726,034 | 224,148,365 |

WADI Scenarios | Total Annual Benefits (EUR) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 29,064,003 | 30,643,655 | 32,706,273 | 35,881,125 | 26,641,243 | 28,170,265 | 32,706,273 | 35,881,125 |

FS1 | 20,626,258 | 22,151,608 | 33,275,869 | 40,903,212 | 19,075,077 | 20,587,187 | 33,275,869 | 40,903,212 |

FS2 | 32,942,354 | 46,373,935 | 46,548,636 | 46,988,116 | 30,191,272 | 44,417,804 | 46,548,636 | 46,988,116 |

FS3 | 33,406,833 | 48,219,300 | 48,394,001 | 54,915,529 | 30,710,414 | 46,185,328 | 48,394,001 | 54,915,529 |

FS4 | 38,614,195 | 51,370,227 | 51,544,928 | 75,236,645 | 35,390,899 | 47,823,846 | 51,544,928 | 75,236,645 |

WADI Scenarios | Total Annual Penalties (EUR) | |||||||

Baseline | 2,470,856 | 2,611,203 | 432,349 | 459,697 | 2,257,577 | 2,385,810 | 118,146 | 581,477 |

FS1 | 3,245,851 | 3,292,577 | 326,846 | 506,598 | 2,452,467 | 2,586,774 | 54,110 | 382,445 |

FS2 | 2,214,386 | 2,724,846 | 459,945 | 469,950 | 2,298,278 | 3,527,484 | 125,687 | 594,446 |

FS3 | 2,235,536 | 2,696,322 | 452,675 | 469,307 | 2,485,131 | 3,675,421 | 123,700 | 593,634 |

FS4 | 3,146,448 | 3,283,159 | 456,063 | 467,608 | 2,874,068 | 3,650,108 | 124,626 | 591,484 |

WADI Scenarios | Total Annual Benefits (EUR) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 11,027,398 | 15,924,814 | 16,711,593 | 21,878,560 | 7,346,406 | 10,657,147 | 11,202,510 | 14,950,226 |

FS1 | 9,373,289 | 13,536,092 | 14,204,854 | 18,596,776 | 6,161,067 | 9,059,977 | 9,513,407 | 12,692,143 |

FS2 | 15,438,358 | 22,294,739 | 23,396,230 | 30,629,985 | 10,261,288 | 14,920,853 | 15,683,515 | 20,954,784 |

FS3 | 11,027,398 | 15,924,814 | 16,711,593 | 21,878,560 | 7,346,676 | 10,657,666 | 11,203,456 | 14,967,703 |

FS4 | 16,541,098 | 23,887,221 | 25,067,389 | 32,817,841 | 11,000,389 | 15,986,499 | 16,803,766 | 22,427,814 |

WADI Scenarios | Total Annual Penalties (EUR) | |||||||

Baseline | 18,854,936 | 20,501,510 | 7,598,174 | 8,375,765 | 4,805,658 | 5,564,719 | 1,655,029 | 2,279,614 |

FS1 | 16,003,268 | 17,401,182 | 6,458,448 | 7,119,400 | 4,065,699 | 4,711,214 | 1,401,002 | 1,930,522 |

FS2 | 26,396,911 | 28,702,114 | 10,637,443 | 11,726,070 | 6,727,922 | 7,830,586 | 2,341,889 | 3,210,311 |

FS3 | 18,854,936 | 20,501,510 | 7,598,174 | 8,375,765 | 4,807,341 | 5,593,869 | 1,655,633 | 2,293,079 |

FS4 | 28,282,404 | 30,752,265 | 11,397,261 | 12,563,647 | 7,208,488 | 8,354,489 | 2,689,422 | 3,425,385 |

^{3}) and of annual hydropower production at HPS Ladhon and Flokas (MWh) for the two different solution processes of the model are presented for the baseline scenario and the four future WADI scenarios. For the irrigation at Flokas the range of the optimized target of the annual water volume for the upper-bound model (f

^{+}) is the same for both solution methods and is equal to its maximum possible value of 174,700,800 m

^{3}. For the lower-bound model (f

^{−}) higher maximum allocation targets are computed by the first solution method (160,079,569, 174,700,800) with much wider ranges between the minimum and the maximum value (14,621,231 m

^{3}) compared to the ones from the second solution method (160,574,718, 160,599,895) with corresponding range (25,177 m

^{3}). By comparing the corresponding results of the FBISP method in this paper with the ITSP as presented in [38], it is worth noticing that the monthly optimized water allocation target values for irrigation at Flokas are equal to the maximum possible allocation for the upper- and the lower-bound solution of the ITSP. This shows that the incorporation of the fuzzy nature of the uncertainties in the FBISP results in lower optimized water allocation target values for the lower-bound (second solution method) solution, reflecting a more analytic and fine approximation of the effect of the uncertainties on the minimum and maximum values of the boundaries of the results.

WADI scenarios | Total Annual Benefits (EUR) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 79,312,739 | 79,312,739 | 86,996,160 | 86,996,160 | 79,312,739 | 79,312,739 | 86,996,160 | 86,996,160 |

FS1 | 67,415,828 | 67,415,828 | 73,946,736 | 73,946,736 | 67,415,828 | 67,415,828 | 73,946,736 | 73,946,736 |

FS2 | 111,037,834 | 111,037,834 | 121,794,624 | 121,794,624 | 111,037,834 | 111,037,834 | 121,794,624 | 121,794,624 |

FS3 | 79,312,739 | 79,312,739 | 86,996,160 | 86,996,160 | 79,312,739 | 79,312,739 | 86,996,160 | 86,996,160 |

FS4 | 118,969,108 | 118,969,108 | 130,494,240 | 130,494,240 | 118,969,108 | 118,969,108 | 130,494,240 | 130,494,240 |

WADI scenarios | Total Annual Penalties (EUR) | |||||||

Baseline | 1,294,736 | 1,295,664 | 235,104 | 244,521 | 1,279,939 | 1,284,407 | 196,202 | 199,798 |

FS1 | 1,071,046 | 1,071,835 | 199,839 | 207,843 | 1,064,610 | 1,070,039 | 168,108 | 172,620 |

FS2 | 1,812,630 | 1,813,930 | 329,146 | 342,330 | 1,791,914 | 1,821,487 | 272,398 | 279,717 |

FS3 | 1,294,736 | 1,295,664 | 235,104 | 244,521 | 1,281,003 | 1,301,062 | 194,570 | 199,372 |

FS4 | 1,942,104 | 1,943,496 | 352,657 | 366,782 | 1,919,908 | 1,934,281 | 292,242 | 299,697 |

Irrigation (m^{3}) | Optimized Target for Total Annual Water Volumes for Irrigation (m^{3}) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 160,079,569 | 174,700,800 | 174,700,800 | 174,700,800 | 160,574,718 | 160,599,895 | 174,700,800 | 174,700,800 |

FS1 | 150,527,097 | 162,767,927 | 174,700,800 | 174,700,800 | 151,272,706 | 151,272,706 | 174,700,800 | 174,700,800 |

FS2 | 160,079,569 | 174,700,800 | 174,700,800 | 174,700,800 | 167,331,623 | 167,331,623 | 174,700,800 | 174,700,800 |

FS3 | 160,541,634 | 174,700,800 | 174,700,800 | 174,700,800 | 167,331,623 | 167,331,623 | 174,700,800 | 174,700,800 |

FS4 | 160,079,569 | 174,700,800 | 174,700,800 | 174,700,800 | 162,606,691 | 162,640,203 | 174,700,800 | 174,700,800 |

^{3}from the first solution method and [(45,840,807, 46,302,872), (5,973,779, 27,652,026),] in m

^{3}from the second solution method.

Hydroscenarios | Annual Shortage for Irrigation (m^{3} × 10^{6}) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Optimistic Solution Method | Pessimistic Solution Method | Best/Worst Case Solution | ||||||||||

(Exceedance Probability %) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) |

1 (68.1%) | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.45 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.45 |

2–18 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

19 (96.9%) | 21.86 | 21.86 | 60.22 | 60.22 | 45.84 | 46.30 | 5.97 | 27.65 | 5.97 | 27.65 | 45.84 | 60.22 |

20–28 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

29 (79.8%) | 0.00 | 0.00 | 0.00 | 0.00 | 9.23 | 9.68 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 9.68 |

30–40 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

41 (93.5%) | 0.00 | 0.00 | 11.83 | 11.83 | 8.67 | 9.11 | 0.00 | 0.00 | 0.00 | 0.00 | 8.67 | 11.83 |

42–43 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

44 (92.9%) | 0.00 | 0.00 | 0.00 | 0.00 | 2.08 | 2.53 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 2.53 |

45–50 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Hydroscenarios | Annual Allocation for Irrigation (m^{3} × 10^{6}) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Optimistic Solution Method | Pessimistic Solution Method | Best/Worst Case Solution | ||||||||||

(Exceedance Probability %) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) |

1 (68.1%) | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.60 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.75 |

2–18 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.61 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.75 |

19 (96.9%) | 1.53 | 1.53 | 1.14 | 1.14 | 1.14 | 1.14 | 1.47 | 1.69 | 1.46 | 1.69 | 1.14 | 1.14 |

20–28 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.61 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.75 |

29 (79.8%) | 1.75 | 1.75 | 1.75 | 1.75 | 1.51 | 1.51 | 1.75 | 1.75 | 1.75 | 1.75 | 1.51 | 1.75 |

30–40 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.61 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.75 |

41 (93.5%) | 1.75 | 1.75 | 1.63 | 1.63 | 1.51 | 1.51 | 1.75 | 1.75 | 1.75 | 1.75 | 1.51 | 1.63 |

42–43 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.61 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.75 |

44 (92.9%) | 1.75 | 1.75 | 1.75 | 1.75 | 1.58 | 1.58 | 1.75 | 1.75 | 1.75 | 1.75 | 1.58 | 1.75 |

45–50 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.61 | 1.75 | 1.75 | 1.75 | 1.75 | 1.60 | 1.75 |

**Table 15.**Annual target, shortage and allocation for irrigation (m

^{3}) for the hydrologic scenario 19.

Baseline | Annual Water Volumes for Irrigation (m^{3}) for the Hydrologic Scenario 19 | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Target | 174,700,800 | 160,079,569 | 174,700,800 | 174,700,800 | 160,574,718 | 160,599,895 | 174,700,800 | 174,700,800 |

Shortage | 60,217,737 | 60,217,737 | 21,860,788 | 21,860,788 | 45,840,807 | 46,302,872 | 5,973,779 | 27,652,026 |

Allocation | 114,483,063 | 114,483,063 | 152,840,012 | 152,840,012 | 114,297,023 | 114,297,023 | 147,048,774 | 168,727,021 |

Shortage/target | 34.5% | 37.6% | 12.5% | 12.5% | 28.5% | 28.8% | 3.4% | 15.8% |

HP Flokas (MWh) | Optimized Target for Total Annual Hydropower Production at HPS Flokas (MWh) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 |

FS1 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 |

FS2 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 |

FS3 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 |

FS4 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 | 19,828 |

^{3}, which are derived from the addition of the optimum target water volumes allocated to the three water users (hydropower production at Ladhon, hydropower production at Flokas and irrigation at Flokas), and the system’s net benefits for these four options for both types of solution methods is shown.

**Table 17.**Maximum allowable (THydroFlokasPlus) and Optimized (Optimized THydroFlokas) monthly targets of hydropower production at Flokas HPS (MWh).

Monthly Hydropower Targets | Maximum and Optimized Monthly Targets of Hydropower Production at Flokas HPS (MWh) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

January | February | March | April | May | June | July | August | September | October | November | December | |

THydroFlokasPlus | 2379 | 2894 | 3435 | 2840 | 1861 | 773 | 251 | 255 | 571 | 1111 | 1397 | 2097 |

Optimized THydroFlokas | 2379 | 2894 | 3435 | 2840 | 1828 | 770 | 251 | 255 | 571 | 1111 | 1397 | 2097 |

WADI Scenarios | Optimized Annual Target for Hydropower Production at HPS Ladhon (MWh) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 275,685 | 289,542 | 278,527 | 291,714 | 179,672 | 179,743 | 186,709 | 199,336 |

FS1 | 275,685 | 289,542 | 278,527 | 291,714 | 179,486 | 179,507 | 186,537 | 199,092 |

FS2 | 275,685 | 289,542 | 278,527 | 291,714 | 179,672 | 180,006 | 186,709 | 199,569 |

FS3 | 275,685 | 289,542 | 278,527 | 291,714 | 179,688 | 180,006 | 186,724 | 199,569 |

FS4 | 275,685 | 289,542 | 278,527 | 291,714 | 179,672 | 179,790 | 186,709 | 199,358 |

**Table 19.**Maximum allowable (THydro-LadhonPlus) and minimum (MinOptimized THydroFlokas) and maximum (MaxOptimized THydroFlokas) optimized monthly targets of hydropower production at Flokas HPS (MWh) and their ratios in (%) for the first solution method (optimistic).

Monthly Hydropower Targets | Target for Hydropower Production at Ladon HPS (MWh) from the First Solution Method (Optimistic) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

January | February | March | April | May | June | July | August | September | October | November | December | |

(1) Thydro-Ladhon Plus | 37,353 | 38,947 | 48,311 | 35,391 | 23,237 | 15,868 | 15,598 | 14,233 | 13,642 | 17,062 | 17,971 | 24,276 |

(2) MinOptimized ThydroFlokas | 37,353 | 38,947 | 48,311 | 35,391 | 23,237 | 15,868 | 12,997 | 14,233 | 6132 | 7387 | 14,395 | 24,276 |

(3) MaxOptimized ThydroFlokas | 37,353 | 38,947 | 48,311 | 35,391 | 23,237 | 15,868 | 14,431 | 14,233 | 12,638 | 11,980 | 15,048 | 24,276 |

(2)/(1) % | 100% | 100% | 100% | 100% | 100% | 100% | 93% | 100% | 93% | 70% | 84% | 100% |

(3)/(1) % | 100% | 100% | 100% | 100% | 100% | 100% | 83% | 100% | 45% | 43% | 80% | 100% |

**Table 20.**Maximum allowable (THydro-LadhonPlus) and minimum (MinOptimized THydroFlokas) and maximum (MaxOptimized THydroFlokas) optimized monthly targets of hydropower production at Flokas HPS (MWh) and their ratios in (%) for the second solution method (pessimistic).

Monthly Hydropower Targets | Target for Hydropower Production at Ladon HPS (MWh) from the Second Solution Method (Pessimistic) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

January | February | March | April | May | June | July | August | September | October | November | December | |

(1) Thydro-LadhonPlus | 20,519 | 30,385 | 27,639 | 17,727 | 11,081 | 10,513 | 13,635 | 11,413 | 5925 | 7387 | 9427 | 13,358 |

(2) Min Optimized ThydroFlokas | 20,519 | 30,385 | 27,639 | 18,390 | 11,081 | 11,247 | 13,635 | 11,413 | 5925 | 7387 | 9427 | 13,358 |

(3)Max Optimized ThydroFlokas | 37,353 | 38,947 | 48,311 | 35,391 | 23,237 | 15,868 | 14,431 | 14,233 | 12638 | 11,980 | 15,048 | 24,276 |

(2)/(1)-% | 55% | 78% | 57% | 50% | 48% | 66% | 94% | 80% | 47% | 62% | 63% | 55% |

(3)/(1)-% | 55% | 78% | 57% | 52% | 48% | 71% | 94% | 80% | 47% | 62% | 63% | 55% |

**Figure 3.**Interconnections between total net benefit and optimized total target for the four options and for both solution methods.

**Table 21.**Interconnections between total net benefit and optimized total target for the four options and for both solution methods.

Baseline | 1st Solution Method | 2nd Solution Method | ||||||
---|---|---|---|---|---|---|---|---|

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Total Net Benefit EUR | 96,192,950 | 102,180,847 | 127,801,604 | 135,950,230 | 104,523,859 | 109,324,450 | 128,786,579 | 134,978,247 |

Optimized target m^{3} | 1,621,866,797 | 1,661,184,353 | 1,641,552,336 | 1,665,055,450 | 1,451,246,461 | 1,451,397,999 | 1,477,912,994 | 1,500,418,501 |

#### 5.2. Results Analysis for the Baseline and the Four Future Scenarios

^{+}) and lower-(f

^{−}) intervals of both solution methods up to 20%.

^{−}) interval for both solution methods of the World Market scenario (FS1) (from 65.5% up to 69.80% farmer income reduction compared to the baseline scenario). The corresponding upper-bound intervals for this scenario drive to a small increase of agricultural income ranging from 1.2% to 11.4% compared to the one for the baseline scenario. These results are similar and compatible with the corresponding ones from the application of the ITSP method in [38]. More precisely, the highest increase of the total system benefits is also observed for the Local Stewardship scenario ranging from 52% to 59% and the only decrease occurs for the World Market scenario (9%–24%). The above mentioned reduction of the agricultural income can be explained by the fact that in the World Market scenario the highest decrease of selling prices and a significant increase of the prices of most of input variables for agriculture (pesticides, seeds, water price, etc.) in comparison to the other scenarios is noticed. Moreover, for most of the crops cultivated at Flokas Irrigation scheme, the presence of area subsidy plays a balancing role for the positive sign (profit) of the agricultural income. In this scenario, no subsidies are provided to the farmers. This fact in combination with the existence of mainly small agricultural units, mainly family farms, renders this agricultural region and Greece in general into weak competitor to bigger and stronger economically agricultural markets, such as U.S.A. or Brazil. Through this analysis, the importance of a balancing area subsidy for economically sensitive agricultural products for the Greek market is verified. The globalization and the liberalization of the agricultural market in combination with the different orientation of the new CAP reform 2014–2020 pose great challenges for the Greek farmers for modernization and increased agricultural and technical expertise.

**Table 22.**Optimized total annual water allocation target of the four future scenarios as ratio of the baseline (%).

WADI Scenarios | Optimized Total Annual Water Allocation Target (m^{3}) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 1,621,866,797 | 1,661,184,353 | 1,641,552,336 | 1,665,055,450 | 1,451,246,461 | 1,451,397,999 | 1,477,912,994 | 1,500,418,501 |

FS1 | 1,612,314,325 | 1,649,251,480 | 1,641,552,336 | 1,665,055,450 | 1,441,612,922 | 1,441,650,080 | 1,477,608,041 | 1,499,983,803 |

FS2 | 1,621,866,797 | 1,661,184,353 | 1,641,552,336 | 1,665,055,450 | 1,458,003,367 | 1,458,598,441 | 1,477,912,994 | 1,500,833,807 |

FS3 | 1,622,328,862 | 1,661,184,353 | 1,641,552,336 | 1,665,055,450 | 1,458,031,456 | 1,458,598,441 | 1,477,941,083 | 1,500,833,807 |

FS4 | 1,621,866,797 | 1,661,184,353 | 1,641,552,336 | 1,665,055,450 | 1,453,278,435 | 1,453,522,032 | 1,477,912,994 | 1,500,457,705 |

FS1/Baseline | 99.4% | 99.3% | 100.0% | 100.0% | 99.3% | 99.3% | 100.0% | 100.0% |

FS2/Baseline | 100.0% | 100.0% | 100.0% | 100.0% | 100.5% | 100.5% | 100.0% | 100.0% |

FS3/Baseline | 100.0% | 100.0% | 100.0% | 100.0% | 100.5% | 100.5% | 100.0% | 100.0% |

FS4/Baseline | 100.0% | 100.0% | 100.0% | 100.0% | 100.1% | 100.1% | 100.0% | 100.0% |

WADI scenarios | Annual Total Net Benefit of Future Scenarios as Ratio of the Baseline (%) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

FS1/Baseline | 79.7% | 80.1% | 89.2% | 92.7% | 81.2% | 81.4% | 89.3% | 92.8% |

FS2/Baseline | 132.9% | 144.6% | 140.7% | 137.7% | 134.0% | 144.3% | 140.6% | 137.8% |

FS3/Baseline | 104.4% | 117.4% | 112.3% | 114.0% | 103.7% | 115.3% | 112.2% | 114.1% |

FS4/Baseline | 145.5% | 155.8% | 152.1% | 165.9% | 146.2% | 155.1% | 152.0% | 166.1% |

**Figure 4.**Box plots for the four options of total net optimized benefits in EUR for the baseline and the four future scenarios (FS1: Future Scenario 1; FS2: Future Scenario 2; FS3: Future Scenario 3; FS4: Future Scenario 4).

**Table 24.**Annual net benefit (EUR) for irrigation and ratios (%) of annual net benefit of the four future scenarios compared to baseline.

WADI Scenarios | Annual Net Benefit for Irrigation (EUR) | |||||||
---|---|---|---|---|---|---|---|---|

1st Solution Method | 2nd Solution Method | |||||||

Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | Min (f^{−}) | Max (f^{−}) | Min (f^{+}) | Max (f^{+}) | |

Baseline | 26,452,801 | 28,172,800 | 32,246,577 | 35,448,777 | 24,255,433 | 25,835,452 | 32,124,796 | 35,762,980 |

FS1 | 17,333,682 | 18,905,757 | 32,769,271 | 40,576,366 | 16,587,306 | 18,037,122 | 32,904,418 | 40,849,102 |

FS2 | 30,217,508 | 44,159,549 | 46,078,686 | 46,528,170 | 27,702,908 | 40,891,389 | 45,954,190 | 46,862,429 |

FS3 | 30,710,512 | 45,983,764 | 47,924,693 | 54,462,854 | 28,162,550 | 42,625,668 | 47,800,367 | 54,791,829 |

FS4 | 35,331,036 | 48,223,779 | 51,077,320 | 74,780,583 | 32,392,287 | 44,325,729 | 50,953,444 | 75,112,020 |

FS1/Baseline | 65.5% | 67.1% | 101.6% | 114.5% | 68.4% | 69.8% | 102.4% | 114.2% |

FS2/Baseline | 114.2% | 156.7% | 142.9% | 131.3% | 114.2% | 158.3% | 143.0% | 131.0% |

FS3/Baseline | 116.1% | 163.2% | 148.6% | 153.6% | 116.1% | 165.0% | 148.8% | 153.2% |

FS4/Baseline | 133.6% | 171.2% | 158.4% | 211.0% | 133.5% | 171.6% | 158.6% | 210.0% |

**Figure 5.**Box plots for the four options of annual net optimized benefits for irrigation in EUR for the baseline and the four future scenarios (FS1: Future Scenario 1; FS2: Future Scenario 2; FS3: Future Scenario 3; FS4: Future Scenario 4).

## 6. Discussion and Conclusions

^{+}) the same for both solution methods and equal to its maximum possible value; for the lower-bound model (f

^{−}) are higher for the first solution method with much wider ranges between the minimum and the maximum value compared to the ones from the second solution method; (ii) for the hydropower production at Ladhon equal to the maximum allowable values for all months except for July, September-November for the first solution method; deviate from the maximum allowable values for all months for the second solution method; and (iii) the maximum possible allocation for all months except May and June for the hydropower production at Flokas. From the optimized targets of the three main users, as analyzed above, it can be concluded that the highest priority for water allocation is set to irrigation, since it has the highest unit benefit, but at the same time also the highest unit penalty. The next priorities are given to hydropower production at Flokas and finally to the hydropower production at Ladhon, which has the smallest unit benefit.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Bekri, E.; Disse, M.; Yannopoulos, P. Optimizing Water Allocation under Uncertain System Conditions for Water and Agriculture Future Scenarios in Alfeios River Basin (Greece)—Part B: Fuzzy-Boundary Intervals Combined with Multi-Stage Stochastic Programming Model. *Water* **2015**, *7*, 6427-6466.
https://doi.org/10.3390/w7116427

**AMA Style**

Bekri E, Disse M, Yannopoulos P. Optimizing Water Allocation under Uncertain System Conditions for Water and Agriculture Future Scenarios in Alfeios River Basin (Greece)—Part B: Fuzzy-Boundary Intervals Combined with Multi-Stage Stochastic Programming Model. *Water*. 2015; 7(11):6427-6466.
https://doi.org/10.3390/w7116427

**Chicago/Turabian Style**

Bekri, Eleni, Markus Disse, and Panayotis Yannopoulos. 2015. "Optimizing Water Allocation under Uncertain System Conditions for Water and Agriculture Future Scenarios in Alfeios River Basin (Greece)—Part B: Fuzzy-Boundary Intervals Combined with Multi-Stage Stochastic Programming Model" *Water* 7, no. 11: 6427-6466.
https://doi.org/10.3390/w7116427