Multi-criteria optimization approach to the design of small dams’ systems along mountainous stream beds

b_i(DH_i)

Thickness (m) of the assumed orthogonal prismatic stone dam, which depends on dam height (DH_i; 0.5, 0.75 or 1 m for heights 1, 1.5 and 2 m, respectively)

BL

Borrow-pit location; BL = 1, 2, 3 and 4 means borrow-pit in A, B, C and D, respectively

bn

Nr of possible borrow-pit's locations (bn = 4)

BP

Borrow-pit

C1

Constraint 1 (further varies as C1a and C1b)

C1a

Constraint 1a: DL_i = 0 (the s/n of the location of dam i equals zero)

C1b

Constraint 1b: DL_i > 61 (the s/n of the location of dam i is larger than 61)

C2

Constraint 2: DL_i = DL_k (the s/n of the proposed location of dam k is the same as dam's i, meaning the algorithm proposes more than one dam at one of the pre-defined possible locations for dam construction)

C3

Constraint 3: DH_i = 0 (the s/n of the proposed height of dam i equals zero)

C4

Constraint 4: BL = 0 (the s/n of the location where the borrow-pit BP is proposed to be placed equals zero)

c_i,agr

Coefficient for dam locations in areas of agricultural land use (see Figure 3 for values)

c_i,constr

Indicatively realistic cost of stonework construction (=70/m³)

c_i,inf

Coefficient for infiltration (see Figure 2 for values)

c_i,settle

Coefficient for locations upstream of settlements (see Figure 3 for values)

CRP

Crossover probability (CRP = 0.4)

Ct_i,j

Indicatively realistic value of stonework transport cost (VB4-lo: Ct_i,j = 0.25 or 0.21 €/m³/km for D_i,j< or ≥5 km, respectively; VB4-hi: Ct_i,j = 2.5 or 2.1 €/m³/km for D_i,j< or ≥5 km)

DH_i

Dam height of ith dam (i = 1–10); can get values 1, 2 and 3 meaning a height of 1, 1.5 and 2 m, respectively

D_i,j

Linear distance (m) of dam i (i = 1–10) from the borrow-pit in location j (j = 1, 2, 3 and 4 for BP in location A, B, C and D, respectively)

DLi

Dam lof ith dam (i = 1–10); can get integer values from 1 to 61

dn

Number of possible dam locations’ values (dn = 6)

E_i(DH_i)

Area (m²) of the cross-section where dam i (i = 1–10) is placed, depending on the value of DH_i

FV

Fitness value; the value of the objective function to be minimized

GAs

Genetic algorithms (optimization method)

hn

Number of possible dam heights’ values; here hn = 3

Li (i = 1–61)

Possible location of a dam; can get integer values from 1 to 61; only used in Tables 4 and 7 

MP

Mutation probability (MP = 0.01–0.04, step 0.005)

NG

Number of generations (NG = 1,500)

PS

Population size (PS = 60)

SC

Selection constant (SC = 3)

SL

Total length of a typical binary chromosome; here, SL = 82

SL₁

Length of the part of a chromosome with all DL_i variables (i = 1–10); here, SL₁ = 60

SL₂

Length of the part of a chromosome with all DH_i variables (i = 1–10); here, SL₂ = 20

SL₃

Length of the part of a chromosome that represents a BL value; here, SL₃ = 2

Soli (i = 1–6)

Solution i (out of 6) for the VB4-lo case

Soli′ (i = 1′–8′)

Solution i′ (out of 8; 1′ to 8′) for the VB4-hi case

VB4-hi

High-cost stonework transport case

VB4-lo

Low-cost stonework transport case

VBi

VB1: Flood protection benefit; VB2: Aquifer recharge benefit; VB3: Dam construction cost; VB4: Stonework transport cost

V_i(DH_i)

Volume of water stored upstream of dam i (i = 1–10), depending on the value of dam i height DH_i

Δ(FV)

Comparison of the fitness value of a solution with the fitness value of the best solution

Mountain floods, especially flash floods, are a natural hazard responsible for many deaths in Europe (Barredo 2009). In the Mediterranean area specifically, floods are one of the most lethal and destructive natural hazards (Gaume et al. 2009), while Greece experienced many flood events during the last decade (Diakakis & Deligiannakis 2017). Current research addresses water resources management of mountainous basins, aiming to create a tool for the optimal design of a small dams’ system in order to mitigate flood effects with the minimum cost, also considering possible simultaneous recharge of underlying aquifers.

River restoration/regulation and, generally, optimal torrential systems’ management is a difficult task as the quantification of the proposed solutions’ costs and benefits is complex. A flood incident can be evaluated through the hydrological intensity of the storm considering the return respective period, but data availability problems can raise accuracy issues (Drobinski et al. 2014), while climate change adds to the uncertainties. Many flood classification scales and indexes have also used that function on a post-flood basis (e.g. Gaume et al. 2009; Borga et al. 2019), considering overflowing of rivers, inundation and damage to roads, vehicles and buildings (Schroeder et al. 2016) or flood intensity described by the return period and inundation duration, combined with flood extent, fatalities, economic and social impacts (Boudou et al. 2016). As far as the efficiency of a torrential system's regulation/management is concerned, it can be assessed by the difference between regulation costs and the expected profit. But while the construction cost can be accurately calculated, there are multiple, long-term and often imponderable benefits from a torrent basin's regulation (Diakakis et al. 2020). While the dam construction costs and material transportation costs can be analytically calculated, the evaluation of the benefit from the dams’ flood protection is extremely complex.

The fact that small dams can be multifunctional makes the evaluation of their benefits even more complex. For example, they are also used for the artificial recharge of underlying aquifers. The site-selection of those dams usually entails large-scale GIS-based and multi-criteria decision analysis techniques. Such criteria are described in Standen et al. (2020), while a guide including simple rules and good practices for the design and construction of recharge dams can be found in CGWB (2007). In general, hydrogeological, meteorological, river water quality, sediment input and other data and parameters, even social considerations, render the optimization of small dams’ systems an extremely complex and challenging task.

Optimization in water resources management problems is generally difficult; these are complex constrained, non-linear, stochastic, multi-criteria optimization problems. Overcoming early conventional optimization techniques’ limitations, metaheuristics and evolutionary algorithms opt for global optimum performance, fast and reliable. Α comprehensive overview of various optimization (meta)heuristics and evolutionary techniques can be found in Kumar & Yadav (2022). Genetic algorithms (GAs) are powerful evolutionary algorithms, successfully applied in water resources, able to provide multiple optimal and alternate sub-optimal solutions of similar cost and value to a problem if properly tuned. GAs are used in this paper, as they have been extensively used in surface and groundwater resources management problems (e.g. Chang et al. 2010; Nicklow et al. 2010; Kontos & Katsifarakis 2012) and because of their versatility in the formulation of the objective function. Some of the established and most sophisticated versions of GAs for multi-objective problems are the multi-objective ‘Non-dominated Sorting Genetic Algorithm II’ (NSGA II) by Deb et al. (2002) and its various updated versions and the Multi-Objective Genetic Algorithm (MOGA) by Reddy & Kumar (2006).

The paper aims to describe an efficient and easy-to-use tool for the optimal design of a system of small dams. Both benefits and cost items of a small dam system depend mainly on dam type, number, locations and dam heights. Current research only considers solid body barrier-type stone dams; hence, optimal design entails the definition of only number, location and height of dams, taking into account their efficiency and their cost. This is achieved through a theoretical, yet convincingly realistic, example.

Current research is based on a theoretical problem. While this is not a case study, the study area is loosely modelled after the Portaikos mountain basin in Northern Greece, for added realism. Input data were harvested from previous research by Gatzoyianni (2006), who investigated the possibility of using small dams in Portaikos stream to recharge aquifers during the rainy season in order to sustain its flow during the summer period, increase underground water reserves and mitigate peak floods downstream. Gatzoyianni evaluated the following criteria in order to empirically propose interesting locations along the stream for the construction of small dams: (a) water infiltration rate through surface soil layers; (b) local stream slope; (c) erodibility of the ground layers; (d) sediment transport deposition rates; (e) local landslips; and (f) plant cover and land use at the vicinity of the proposed dam sites.

Building on this research, the current paper discusses the optimal management of surface water resources in a basin loosely modelled after the Portaikos Basin. The actual basin stretches mainly over Trikala regional unit, western Thessaly, Greece. It is a sub-basin of the greater basin of Pinios River that has attracted a lot of scientific interest (e.g. Mimikou & Baltas 2013). A comprehensive presentation of the area is given in the PhD thesis of Stefanidis (2018). Data regarding the geology of the area can be found in Pomonis et al. (2005). As the authors’ aim is to present a general methodology for optimizing small dam systems, without focusing on particular details, the simplified data sets and maps used by Gatzoyianni (2006) are adopted. A topographic map of the actual Portaikos Basin is provided as Supplementary Material (SM1), presenting water basin boundaries, contour lines, hydrographic network, road network and settlements.

The theoretical problem can be stated as follows: (a) how many small dams (up to a pre-defined number) should be constructed; (b) at which locations (from a list of preselected sites); (c) what should their height be (three dam heights are allowed); and (d) where a BP should be built for the respective aggregates’ (stonework) extraction, in order to achieve the following optimization goals: (i) maximization of the flood protection benefit of the downstream area; (ii) maximization of the benefit from the local aquifer's artificial recharge (if applicable); (iii) minimization of dams’ construction cost; including (iv) minimum stonework transportation cost from the BP to the dam sites. The proposed three-stage optimization approach is graphically presented in Figure 1. ‘Expert’ tool means that the expertise of relevant scientists (here, authors) is needed to identify potential DLs and BPs and characterize DLs regarding their importance on the difficult-to-quantify flood protection and aquifer recharge benefits.

The locations of the proposed small dam and BP sites are marked on the gross soil type and land-use maps included in Gatzoyianni (2006), as shown in Figures 2 and 3. In total, the system loosely modelled after Portaiko's theoretical stream system is comprised of five branches progressively conjoined in a central river bed (Figure 2). Its total length is in the order of 50 km.

In order to evaluate a dam system, a plethora of data is required: (a) the volumes of water stored upstream of each dam's potential location depending on its height; (b) the soil type; (c) the land use of each potential location; and (d) the proximity of a potential dam location (DL) to settlements, which adds value to it, relative to other rival locations, assuming it provides protection flood services. To obtain the aforementioned input data, simulation of the hydrographic network through suitable software is needed, based on the respective characteristics of the study area.

The data pre-processing stage includes experts implementing the following steps (see Figure 1):

S1. Use of a topographic map to identify and pre-determine the potential dams’ locations, the potential BP locations and then calculate the distances between all possible dam locations and possible BP locations.

S2. Use of the Digital Elevation Model (DEM; here, the freely provided by Google Earth) and Hydrologic Engineering Center - River Analysis System (HEC-RAS; USACE 2016; see SM2 for all relevant files) not to typically simulate the flow, but rather to approximately pre-calculate the potential dam locations’ cross-section areas and subsequently, face areas of dams and indirectly the respective stonework volumes (S2a) and also pre-calculate the reservoir (max stored water upstream of dams) volumes (S2b).

S3. Use of a geological map to characterize all possible dam locations for their infiltration capacity, hence the locations’ ability to recharge the underlying aquifer.

S4. Use of a land uses map to characterize all possible dam locations for their proximity to (or impact to flood protection of) settlements (S4a) and their proximity to agricultural land (S4b).

S5. Use of respective contemporary legislation/regulations/price catalogs to define current stonework construction costs in €/m³.

S6. Use of respective contemporary legislation/regulations/price catalogs to define current transportation costs (from a BP to a small dam) in €/m³/km.

The 61 potential dam locations selected, combined with three potential dam heights result in 183 stonework and reservoir volumes to be calculated. The simplified conceptual model involves no hydraulic simulation; flow characteristics are not considered, while no dam is assumed to affect another dam's benefit/cost values. Another practical simplification assumption is that any flood incident will result in all new small dams (regardless of the selected construction locations) overflowing. Hence, all partial benefit/cost values can be separately calculated prior to the optimization process, for all 183 combinations of 61 single potential dam locations and 3 heights. Any solution proposing a system of dams of various heights and a BP is assumed to exhibit a total management value equal to the sum of the respective partial benefit/cost values of each dam. This drastically reduces the expected optimization computational load, while there is no need for a complex coupled simulation–optimization model.

GAs is the selected optimization tool in our case; two minimization and two maximization goals were combined. The problem has been formulated as a single-objective minimization one: the sum of benefits minus costs is to be minimized. It is mathematically described as follows:

where FV is the fitness value, DL_i is the serial number (s/n) of the ith out of Ν (Ν = 1–10) locations where a dam will be constructed, which can have integer values from 1 to 61 (61 pre-defined locations; Figure A.1 of the Appendix); DH_i is the s/n of the respective height value of dam i out of Ν (Ν = 1–10), which can have integer values from 1 to 3 (three pre-defined dam heights; DH_i = 1⇒h_i = 1 m, DH_i = 2⇒h_i = 1.5 m, DH_i = 3⇒h_i = 2 m); BL is the s/n of the location of the four pre-defined potential locations where the BP will be placed (locations A, B, C, and D in Figures 2 and 3, BL = 1⇒BP at A, BL = 2⇒BP at B, BL = 3⇒BP at C, BL = 4⇒BP at D); VB1 is the benefit from the flood protection services of the dams; VB2 is the benefit from the aquifer's artificial recharge; VB3 is the dams’ scheme construction cost; and VB4 is the transport cost of the required aggregates from the selected BP location (BL) to the proposed dams’ locations.

Depending on the case study with its local peculiarities and characteristics, as well as the importance given by the local or/and managing authorities to each benefit/cost item, various weighting factors can be assigned to each item in the objective function. In this pilot approach, aiming to present the general methodology, no weighting factors have been used (hence, they can be assumed to be equal to 1).

The values of V_i(DH_i) for all possible combinations of the 61 potential small dam locations and the three height values of each dam have been pre-calculated with the help of HEC-RAS (see all relevant application files in SM2). They are summarized in Figure A.1; the full dataset is included in the Supplementary Material (SM3). The values that were attributed to c_i,settl, based on whether the respective potential DL is upstream of a settlement, are presented in Figure 3. In this way, the final VB1 values are calculated for all possible combinations of dams’ locations–heights (Figure A.2; SM3 for the full dataset). The process of obtaining the crucial value c_i,settl in a case study should be decided more precisely, e.g. by flood-scenario simulations followed by damage cost assessment.

The soil type, the respective geological formation, the infiltration coefficient and the respective value of c_i,settl are shown in Figures 2 and A1. Land uses and the arbitrarily attributed respective enhancement coefficient of c_i,agr are shown in Figures 3 and A.1 and SM3. The calculated final values of VB2 for any combination of DL-height are summarized in Figure A.2 (and SM3).

Values of Ε_i(DH_i), together with the resultant stonework volumes Ε_i·b_i, for any combination of DLs and DHs have been pre-calculated, using HEC-RAS (Figure A.1; SM3 for the full dataset). The calculated final values of VB3 for any combination of DL-height-thickness are summarized in Figure A.2 (and SM3).

The distances D_i,j have been pre-calculated (Figure A.1; SM3). Based on them, the final values of VB4 for any combination of DL-height- and BL have been also pre-calculated. They are shown in Figures A.2 and A.3 of SM3 for VB4-lo and VB4-hi, respectively.

The constraints C1–C4 stated in Equations (2)–(5) are the following: C1a, DL_i = 0 (the s/n of dam site i equals zero); C1b, DL_i > 61 (the s/n of dam site i is larger than 61); C2, DL_i = DL_k (the s/n of the proposed location of dam k is the same as dam's i, meaning the algorithm proposes more than one dam at one of the pre-defined possible dam locations); C3, DH_i = 0 (the s/n of the proposed height of dam i equals zero); and C4, BL = 0 (the s/n of the BL is zero).

Constraints’ violations are handled with the repair method. The repair method was preferred over the use of penalties, for the following reasons: (a) it is closer to the ‘nature’ of the examined problem, since, in essence, it attributes no dam to non-available locations and (b) it entails a lower computational volume; there is no need to investigate the penalty magnitude through time-consuming tests. Table 2 presents the exact pairing of all possible decision variable values, converted from binary format of a chromosome to decimal format (chromo), with the ultimately used values, followed by the respective physical interpretation.

C1a constraint violation is handled with the Repair1a technique: any chromosome/solution that contains at least one DL value equal to zero (DLi = 0) is treated as if no dam is constructed in the respective location (Table 2; No Dam i). While the initial population is allowed to include chromosomes/solutions with one or more dam locations equal to zero, Repair1a is implemented in the evaluation operator. If crossover-mutation operators modify a feasible solution into an infeasible one concerning C1a constraint violation, there is no respective handling applied; the next generation's evaluation and Repair1a will deal with it.

C1b constraint violation is handled with the Repair1b technique: any DL_i > 61 in a chromosome is paired with used value 0 (DL_i = 0); hence, following the Repair1a route, it is treated as ‘no dam in respective location’ (Table 2; No Dam i). Just like C1a, while the initial population allows the C1b violation, Repair1b is implemented in evaluation, with crossover-mutation not participating in handling, even infeasible solutions are produced (due to C1a violation).

C2 constraint is practically trying to handle the fact that variable DL_i can obtain the same value for different values of i, for each one of i = 1 to Ν (Ν = 1–10) dam sites, namely the algorithm proposes more than one dam at a single pre-defined spot. A simple Repair2 rule is used: should a DL_i for a random i=k obtain a value already attributed to one for a different i, then DL_k = 0, namely no kth dam is constructed (Table 2; No Dam k). Just like C1a and C1b, the initial population allows C2 violations, but Repair2 is implemented in evaluation, while crossover-mutation does not take part in handling C2 violations.

C3 violation is handled with the Repair3 technique: any chromosome that contains at least one proposed s/n of DH equal to zero (DH_i = 0) is treated as ‘no dam in respective location’ (Table 2; No Dam i). Again, the initial population allows C3 violations, while Repair3 is implemented in evaluation; crossover-mutation does not participate in handling C3 violations. Handling of C1a, C1b, C2 and C3 regarding the interpretation of DLi = 0, DL_i > 61, DL_i = DL_k and DH_i = 0 conditions as ‘no dam’ conditions is not random. After a first series of tests where it was obvious that the best solutions included a number of dams quite lower than the max allowed (=10), the authors wanted to direct the algorithm to solutions with few dams. The constraint handling selected is actually inserting bias in the optimization procedure towards solutions/strategies with fewer possible dams.

C4 constraint violation is handled with the Repair4 technique: the decimal value of each part of a chromosome representing BL is increased by 1 (BL = BL + 1) so that the initial values {0, 1, 2, and 3} correspond to {1, 2, 3, and 4}, which imply the pit is located at {A, B, C, and D} (Table 2). Hence, value BL = 0 does not need handling as no chromosome-originated value is redundant. ‘init’ allows the violation, ‘eval’ applies Repair4, while ‘cr-m’ do not participate in C4 handling.

The genetic operators used are selection (tournament procedure with elitism), evaluation, crossover and mutation. After an extended series of tests regarding the genetic parameters’ values, population size (PS) is defined as PS = 60; higher PS values do not deliver better solutions. The number of generations is defined as NG = 1,500; more generations do not exhibit improvements to compensate for the added computational time. The selection constant (SC), SC = 3 is used as this is a common and established value in relevant problems and literature. Crossover probability (CRP), CRP = 0.40 is used following previous own experience (Kontos & Katsifarakis 2017a, 2017b). Concerning mutation probability, MP, the results of a series of tests for values from MP = 0.01 (≈0.82/SL) to 0.04 (≈3.28/SL), step 0.005 are presented. This way, both the old empirical rule suggesting MP ≈ 1/SL (here 1/82 ≈ 0.012) and the rule produced by earlier own research suggesting MP ≈ 2–2.5/SL (≈0.024–0.030) (Kontos & Katsifarakis 2012) are tested.

The programming code was written in Visual Basic and a respective executable application was built with a user-friendly interface that can facilitate the reproduction of all simulations and results by any user with no programming skills. The simulation series consists of 10 runs for each 1 of the 7 MP values (0.01–0.04, step 0.005), a total of 70 runs. The simulation series is applied for the VB4-lo and the VB4-hi cases, for a total of 140 runs. All runs’ configurations and results are presented as Supplementary Material (SM4).

For the low-cost stonework transport case (VB4-lo), the 70 runs produced six discrete solutions (Sol1–Sol6) as far as dam – BP layouts are concerned, but with similar FV values. The usefulness of the sub-optimal solutions is to allow consideration of (a) additional, mainly intangible, criteria, not formally included in the optimization process and (b) the sensitivity of optimization results to small changes in input factors.

The 70 runs required about 10 min of simulation time (Intel Core i7 7700 @3.60 GHz; 16 GB RAM @1197 MHz). Table 3 presents the FV, VB1, VB2, VB3 and VB4 values of all solutions as well as the increase (%) in FV that solutions exhibit compared to the best solution Sol4. A green–white–red colour scale is utilized to better present low–medium–high values per column. Table 4 presents the different dam – BP layouts/configurations of these six solutions, including a number of dams, dam locations and respective dam heights, together with the BP's location, as well as the number of appearances of each DL in the six solutions. It is obvious that the six solutions constitute six different strategies, regarding the layout of the dams; no pair of solutions exhibits exactly the same locations. Figure A.4 is a graphical representation of the six identified (sub)optimal solutions for the VB4-lo case, presenting dam locations and heights, BL, together with relevant info, a ranking based on FV, the number of dams proposed and the increase in FV compared to the best solution (Sol4).

Table 3

Results for all 6 solutions produced by all 70 runs for the low-cost stonework transport case (VB4-lo)

Fitness value (FV), flood protection (VB1) and aquifer recharge (VB2) benefits, dam construction (VB3) and stonework transport (VB4) costs, appearance rate and increase of FV compared to the best solution (4) are presented.

^aColour scale (per column): green–white–red = low–medium–high values.

^bComparison with the best solution Sol4 (FV_i-minFV).

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Table 4

Results for all solutions produced by all 70 simulations for the low-cost stonework transport case (VB4-lo)

Nr of dams, dam locations and respective heights, together with the BP's location are presented.

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Table 5 presents the produced solutions (Sols) by the seven MP value tests for each 1 of the 10 runs per MP value, together with the number of different solutions found and the number of times the optimal solution (Sol4) was found. Given that the number of runs per MP value cannot guarantee statistically safe conclusions, there is no clear evidence of the impact of the MP value on the consistency of finding the optimal solution or the ability to identify many sub-optimal solutions. It seems that values 0.02 and 0.025 satisfy both requirements, leaning more towards the first.

In the VB4-lo case, given the weighting factors assigned to the various benefit/cost items in the objective function (practically = 1; see Equation (1)), the importance of VB1 on the final FV value is the highest of all benefit/cost items (≈39–41%), followed by VB3 (≈22–30%) and further away by VB2 (≈30–37%), while VB4's contribution is trivial (≈0.3–0.4%). The best solution is Sol4, proposing the construction of five dams, exhibiting: FV₄ = −6,663.49 (|FV₄| = 25,467.61), VB1₄ = −10,071.60 (39.55% of |FV₄|), VB2₄ = −5,993.95 (23.54% of |FV₄|), VB3₄ = 9,304.65 (36.54% of |FV₄|), VB4₄ = 97.41 (0.38% of |FV₄|). Despite the fact that VB1 is generally and specifically in this solution, the most important benefit/cost item of the objective function, Sol4 exhibits min FV, but is ranked 2nd based on VB1 value (low to high; see Table 3). It is ranked first based on VB2, sixth based on VB3 and fifth based on VB4. It is graphically presented in Figure A.4a. The second best solution is Sol3, again considering five dams, with an FV of only +0.83% compared to the best solution (Figure A.4b). The third best solution is Sol2, again considering five dams, exhibiting an FV of just +1.18% compared to the best solution (Figure A.4c).

Checking data from Figures 2, 3 and A.1 and trying to justify why the specific eight dam locations dominate the produced solutions (in order of a number of appearances L46, L59, L1, L61, L2, L44, L54, L4), one can see that L46 exhibits low to medium stored water volume enhanced by the highest c_i,settl = 2.5, due to settlement proximity (VB1). It is also linked with low infiltration (c_i.inf = 0.05), but with c_i,inf = 0.8 and c_i,agr = 1.2 (VB2), while the respective stonework volume is low to medium (VB3). L59 exhibits the highest stored water volume (VB1), multiplied by c_i,inf = 0.8 and c_i,agr = 1.2 (VB2) while being linked with low stonework volume (VB3). L1 is linked with a low to medium stored water volume upstream (VB1), but multiplied by the highest coefficient c_i,inf = 0.8 for infiltration and enhanced by c_i,agr = 1.2 for the added value of agricultural water use (VB2), while the stonework volume contributing to dam construction cost is low (VB3). L61 exhibits low to medium stored water volume with c_i,setll = 1.2 (VB1), c_i,inf = 0.8 and c_i,agr = 1.2 (VB2) and low stonework volume (VB3). L2 exhibits low to medium stored water volume upstream (VB1), c_i,inf = 0.8 and c_i,agr = 1.2 (VB2) and low stonework volume (VB3). L44 is linked with the second highest stored water volume upstream (VB1), medium infiltration coefficient (c_i.inf = 0.3) and c_i,agr = 1.2 (VB2), but a high stonework volume (VB3). L54 exhibits low to medium stored water volume upstream (VB1), c_i,inf = 0.8 and c_i,agr = 1.2 (VB2) and low stonework volume (VB3). L4 is linked with low stored water volume upstream (VB1), c_i,inf = 0.8 and c_i,agr = 1.2 (VB2) and low stonework volume (VB3).

The cost of stonework transport VB4, though quite small compared to the other benefit/cost items, is well perplexed in the final/(sub)optimal solutions; it always resides in the sweet spot that balances optimal distance from the dam locations and the respective stonework volumes. Specifically, there are two general solution profiles identified in the six solutions of VB4-lo: (a) solutions like Sol2, Sol4 and Sol6 that include a dense group of four dams (two of L1, L2, L44, L61 plus L46, L59), each one of which is beneficial for various reasons, plus a dam in L44 that is distant from the group (high stored water volume upstream; medium infiltration capacity with increased importance due to agricultural and high stonework volume) and a BP at B (between the group of four dams and L44) and (b) solutions like Sol1, Sol3 and Sol5 that include a dense group of dams (three of L1, L2, L4, L54, L61 plus L46, L59) and a BP at D, the closest to all dams. It is important to mention that the best solution Sol4, which belongs to the general profile ‘a’ and the second best solution Sol3, which belongs to the general profile ‘b’, exhibits a difference of only 0.83% in the assigned fitness value (FV).

For the high-cost stonework transport case (VB4-hi), the 70 runs implemented produced eight discrete solutions (Sol1΄–Sol6΄) as far as their FV is concerned. The 70 runs required about 10 min of simulation time. Table 6 presents the FV, VB1, VB2, VB3 and VB4 values of all solutions as well as the increase (%) in FV solutions exhibited compared to the best solution Sol2΄. A green–white–red colour scale is utilized to better present low–medium–high values per column. Table 7 presents the different dam – BP layouts/configurations of these eight solutions, including the number of dams, dam locations and respective dam heights, together with the BP's location, as well as the number of appearances of each DL in the eight solutions. Six out of eight solutions constitute distinct strategies, judging by the dam locations’ layout alone (Sol1΄, Sol3΄–7΄), while Sol2΄ and Sol8΄ can be categorized in the same strategy. They exhibit the exact same layout (L1, L2, L46, L59 and L61), with the only difference being the DH at L2 (2 m for Sol2΄, 1 m for Sol8΄). Hence, the eight solutions of VB4-hi correspond to seven basic management strategies.

Table 6

Results for all 8 solutions produced by all 70 runs for the high-cost stonework transport case (VB4-hi)

Fitness value (FV), flood protection (VB1) and aquifer recharge (VB2) benefits, dam construction (VB3) and stonework transport (VB4) costs, appearance rate and increase of FV compared to the best solution (Sol2) are presented.

^aColour scale (per column): green–white–red = low–medium–high values.

^bComparison with the best solution Sol2΄ (FV_i-minFV).

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Table 7

Results for all solutions produced by all 70 runs for the high-cost stonework transport case (VB4-hi)

Nr of dams, dam locations and heights, together with BP's location are presented.

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Figure A.5 is a graphical representation of all eight identified as (sub)optimal solutions for the VB4-hi case, presenting dam locations and heights, BL, together with relevant info, ranking based on FV, the number of dams proposed and the increase in FV compared to the best solution (Sol2΄). Table 8 presents the produced solutions (Sols) by the seven MP value tests for each 1 of the 10 runs per MP value, together with the number of different solutions found and the number of times the optimal solution (Sol2΄) was found. There is no clear evidence of the impact of the MP value on the consistency of finding the optimal solution or the ability to identify many sub-optimal solutions, nor could a general rule be devised.

Comparing the VB4-lo and VB4-hi cases’ results, there are similar strategies identified (same dam locations, different dam heights or/and BL). Specifically, Sol1 and Sol3΄ share the same dams’ layout (L1, L2, L4, L46 and L59) and propose the construction of the BP at the same location (D), but propose different heights for the dam at L46 (1 and 1.5 m, respectively). Sol2΄and Sol8΄, which have already been identified to belong in the same basic strategy, find another version/variation of Sol3. It exhibits the same dams’ layout (L1, L2, L46, L59 and L61), and the same BL (D), but a different DH at L2 than the others (1 m instead of 2 m). Finally, Sol5 and Sol1΄ both propose dams at L1, L46, L54 and L59, L61, and the BP at D, but differ concerning the DH at L59 (1 and 2 m, respectively).

In the VB4-hi case, given the weighting factors assigned to the various benefit/cost items in the objective function (practically = 1; see Equation (1)), the importance of VB1 on the final FV value is (just like VB4-lo) the highest of all benefit/cost items (≈37–39%), followed by VB3 ( ≈ 28–30%) and VB2 (≈29–31%); VB4's contribution is small, but higher than the respective importance in VB4-lo (≈3%). The best solution is Sol2΄, which is not found in VB4-lo, nor even a similar strategy (at least same dams’ layout); Sol2΄ proposes the construction of five dams, exhibiting: FV_2΄΄ = −6,064.65 (|FV_2΄| = 19,425.97), VB1_2΄ = −7,275.58 (37.45% of |FV_1΄|), VB2_2΄ = −5,469.73 (28.16% of |FV_2΄|), VB3_2΄ = 6,076.40 (31.28% of |FV_2΄|), VB4_2΄ = 604.26 (3.11% of |FV_2΄|). Sol2΄ exhibits min FV and is also ranked first based on VB1 and VB2 values (low to high; see Table 6). However, it is ranked eighth based on VB3 and VB4. Sol2΄ is graphically presented in Figure A.5a. The second best solution is Sol3΄ (same strategy, different version of Sol1 of VB4-lo), again considering five dams, with FV_3΄ just +1.53% compared to the best solution (Figure A.5b).

Six out of eight solutions of VB4-hi suggest the construction of five dams (Sol1΄, Sol2΄, Sol3΄, Sol5, Sol7΄ and Sol8΄; see Table 8), while two of them propose four dams (Sol4΄ and Sol6΄). Actually, Sol4΄ is Sol2΄ without a dam at L61, while Sol6΄ is Sol1΄ without a dam at L54. It is interesting to discuss the different strategies of Sol4΄ and Sol6΄; these solutions propose the use of only four dams. The best of the two, Sol4΄, is the fourth best solution, as far as FV is concerned (only +2.38% compared to the best solution) and exhibits: FV_4΄ = −5,920.40 (|FV_4′| = 16,710.32), VB1_4΄ = −6,417.61 (38.41% of |FV_4΄|), VB2_4΄ = −4,897.75 (29.31% of |FV_4΄|), VB3_4΄ = 4,899.92 (29.32% of |FV_4΄|), VB4_4΄ = 495.04 (2.96% of |FV_4΄|). Sol4΄ exhibits the fourth min FV, but is ranked 7th based on the VB1 value (low to high; see Table 6). It is ranked fifth based on VB2, second based on VB3 and third based on VB4. It is graphically presented in Figure A.5c.

Just like VB4-lo, the algorithm balances flood protection/recharge benefit and construction/transport costs, through certain combinations of groups of five, but also here, at four, dam locations. All solutions are exclusively associated with BL, D, while using only 8 out of 61 possible dam locations (L1, L2, L3, L4, L46, L54, L59 and L61 for 8, 6, 1, 1, 7, 2, 8 and 5 times, respectively; see Table 7 and Figure 6(b)). The only difference between VB4-hi and VB4-lo solutions is that L44, used extensively in VB4-lo (in the best solution too), is replaced by L3, which appears one time in Sol7΄ (6th best solution). Dam locations L1 and L59 are included in every solution (Figure 6(b); deep blue colour), while L2, L4, L46 and L61 are included in most solutions (Figure 6(b); light blue colour). Figure 6(b) presents all the 61 possible dam locations, the eight locations proposed in the eight produced solutions and the BL proposed in the VB4-hi case.

Apart from L1, L2, L4, L46, L54, L59 and L61 whose features were described in the discussion of VB4-lo results, the newcomer L3 exhibits low to medium stored water volume upstream (VB1), high infiltration capacity (c_i,inf = 0.8) and the highest land use coefficient c_i,agr = 1.2 (VB2), together with very low stonework volume (VB3).

The cost of stonework transport VB4, though still quite small compared to the other benefit/cost items, is ten times higher than VB4-lo in this case (VB4-hi), leading to different solutions and strategies. The genetic algorithm responded well and produced solutions with the dams located more densely (the distant L44 used in VB4-lo, is not used now), in groups of four or five close to each other and close to the only proposed BL, D (Figure A.5). Hence, the two identifiable general solution management profiles in VB4-hi are: (a) solutions like Sol1΄, Sol2΄, Sol3΄, Sol5΄, Sol7΄, and Sol8΄, that include a dense group of five dams (three of L2, L3, L4, L46, L56, L61 plus L1 and L59) and (b) solutions like Sol4΄ and Sol6΄, that include a dense group of four dams (one of L2, L61 plus L1, L46 and L59); all with the BP at D.

A novel method to optimize a small dams’ scheme, namely dam locations, dam heights and a BP's location is proposed. Sixty-one possible dam locations (allowing selection of up to 10), 3 possible heights for each dam (allowing selection of 1) and 4 possible BP locations (allowing selection of 1) are considered. Environmental management optimization problems, like the one studied, belong to the family of complex, constrained, non-linear, stochastic and multi-criteria optimization problems. GAs are used as the optimization method. The new feature of this research is the simplifying, yet novel, formulation of the optimization problem into a multi-criteria minimization problem and the use of a genetic algorithm for the optimization. The very optimization problem per se is also something new, as to the authors’ knowledge, it has not been approached with optimization tools up to now.

The initial problem is a multi-objective one; the four objectives are (a) maximization of flood protection gain (VB1); (b) maximization of gain by artificial recharge of underlying aquifers (VB2); (c) minimization of construction cost (VB3); and (d) minimization of stonework transport cost from the BP to the dam locations (VB4). The problem is then converted to a single-objective one: minimize the sum of all four gain/cost items. The simplification of the objective function and the use of simple GAs instead of multi-objective evolutionary algorithms are compensated by sophisticated post-processing of results, including a systematic investigation of proposed solutions, the identification of various alternative good (sub)optimal solutions and their classification in different management strategies.

Flood protection benefit (VB1) is simply represented only by the volume of water stored upstream, locally enhanced by a coefficient when settlements are close downstream. Benefit from the possible artificial recharge of the underlying aquifer (if any) by the stored water in reservoirs upstream of dams (VB2) is represented by the aforementioned stored water volume multiplied by a coefficient regarding land use (agricultural lands provide a bonus) and a coefficient regarding the infiltration capacity of the area of each DL. The dam construction cost (VB3) is represented by the stonework volume needed, multiplied by the cost (€) of stonework construction per m³. Stonework transport cost from the BP to each DL (VB4) is represented by the distance between the pit and the dam, multiplied by the cost (€) of transport per km per m³ of stonework and the stonework volume. Two scenarios regarding the transport cost per distance unit are presented, VB4-lo and VB4-hi (10 times higher) in order to investigate the algorithm's performance, through its response to the variation of the lowest of the cost items. The variation of the value of the less influential to the total management cost VB4 leads the genetic algorithm to readapt and reach the new optimal and sub-optimal solutions; this is a strong indication that the configuration and performance of the algorithm and the proposed methodology are good.

The four objectives (benefits VB1, VB2 and costs VB3, VB4) are assumed to be assigned a weighting factor equal to 1 in the current pilot implementation. The VB1 and VB2 benefit (negative cost) items are also practically expressed in €. Given the current objective function with the assigned contribution of each cost item and the specific studied problem, all optimal solutions entail negative costs. This means there is an overall benefit from the implementation of the management scheme application.

Four constraints of the optimization problem originate from the binary genetic encoding of solutions as chromosomes in the simple binary genetic algorithm used. They are related to the infeasible values that the dam locations, dam heights and BL of each solution can obtain. In this paper, all constraints are handled with the repair method (repairing infeasible solutions/chromosomes); the infeasible DL or/and height values are interpreted as a ‘no dam in that location’ order. In this way, the algorithm is allowed to investigate and propose solutions with smaller than the max allowed number of dams (dn = 10).

Concluding on the results of the two VB4 scenarios, in VB4-lo, the algorithm manages to produce six different management solutions. Two solutions dominate the 70 runs (41% + 41% = 82%, respectively), while VB4-lo solutions exhibit higher variation regarding the dam locations (compared to VB4-hi) and propose two locations for the BP. Two general solution profiles are identified in VB4-lo: (a) a dense group of four dams plus a distant dam, with the BP between the group and the distant dam and (b) a dense group of dams alone, with the BP close to them. On the other hand, VB4-hi, with a 10-fold transport cost, produces 8 different solutions; 1 dominates the runs, appearing in 54% of them. Two general solution profiles are identified in VB4-hi: (a) a dense group of five dams, with the BP at D and (b) a dense group of four dams, with the pit also at D. Profile ‘b’ solutions exhibit a very small increase in the total management cost (+2.38% to +3.39%) compared to ‘a’.

It is observed that the implemented genetic algorithm responded to the large increase of the otherwise low transport cost in VB4-hi, adapting to the new conditions and producing solutions with more spatially concentrated dams, closer to the accordingly selected BL. On the contrary, in VB4-lo, the algorithm adapted to the lower importance of the transport cost over the total ‘cost’, leading to solutions proposing dams exhibiting higher flood protection benefits, albeit in larger distances amongst them.

MP value variation, ranging within the limits stated by relevant literature, does not seem to have a great impact on the appearance rate of the optimal solutions in either scenario. Values around 0.02, though, seem to generally be able to provide a high rate of optimal solutions’ appearance, combined with a satisfactory number of alternate sub-optimal solutions/strategies in the studied problem formulation.

In general, the genetic algorithm, as configured and implemented by the research team, managed to consistently (with a high success rate) produce the optimal solutions for both transport scenarios. Of course, one should bear in mind that (a) in such complex problems, no one can guarantee/prove there are not any better solutions until they are found, (b) even the smallest modification in the objective function formulation (e.g. different weighting factors in the cost/benefit items) can lead to different, perhaps better, solutions, (c) the use of a computationally demanding optimization tool, like GAs, renders the simplification of the conceptual and simulation models imperative; in order to balance between accuracy and computational efficiency, many assumptions and simplifications are made, which makes the myopic search for an algebraically optimal solution less important than the more practical search for many alternatives (sub)optimal solutions. The creation of a pool of various good management solutions and strategies for such complex and scientifically and even socio-economically volatile environmental resource management problems is more important. Such an approach can act like a decision-support mechanism for the management authorities in the design phase, providing alternatives, even if some initial data are modified (a-posteriori modified constraints, added constraints, modified weighting factors, budget constraints, etc.).

Towards this direction, the proposed methodology/algorithm managed to successfully identify the algebraically best solutions in both scenarios, but also to identify various alternate solutions/strategies, which are actually local minima of the objective function. This way, if for example, after the completion of a similar real dams’ scheme management case study, the construction of a dam in a certain location is not admissible due to new circumstances (false data, physical disasters, socio-economic reasons, etc.) or the construction of the BP is forbidden in a certain location, another solution/strategy may be selected from the pool and serve as an alternative. As proved, the alternatives produced are very close to the overall optimal solutions, concerning the benefits and costs and the total FV, which renders all proposed solutions perfect candidates to fill the alternative solutions’ pool.

Overall, this pilot attempt to propose a novel methodology to improve water resources management in mountainous stream beds, optimizing a small dams’ scheme, is promising. The ‘OptiDams’ software application created is an efficient optimization tool that can consistently produce optimal and alternative sub-optimal management solutions and strategies, through a user-friendly interface. The simulations, though based on simplifying assumptions, pave the way to a gradual increase in realism towards the creation of a widely applicable, fully operational software application for real-world problems.

The novel small dams’ management optimization methodology proposed is a decisive first step towards the use of evolutionary algorithms for the realistic solution of similar complex multi-criteria integrated water resources management problems; a complexity that mostly originates from the need to quantify benefits, losses, risks, disasters and costs of complex situations and conditions, such as flooding and flood protection, infiltration benefits from artificial reservoirs upstream dams, dam construction costs, construction material transport costs, etc. The weaknesses of this first approach is actually the ground where future research could and should flourish.

Firstly, given the currently proposed formulation of the objective function and simplified conceptual model, tests of the response of the algorithm to various weighting factors of each benefit/cost item should be made and general conclusions are drawn, directing potential tool users towards better applications.

Moreover, other constraint handling techniques should be investigated, in order to propose the most suitable to produce the optimal and alternate sub-optimal solutions and management strategies. The different constraint handling approaches must be evaluated by the ability or not to find the optimal solution, the consistency in finding (appearance rate of) the optimal solution in a specific number of runs, the time needed for the convergence to the optimal solution and the variety of the proposed (sub)optimal solutions and respective management strategies.

Next, the conceptual model itself should be refined. The simulation/calculation and simplifying assumptions of flood protection benefits should be further investigated and better represented. Instead of the volume of water stored upstream of a dam, depending on the respective topography and DH, the flood protection benefit/importance could be better quantified through the actual calculation of the flooding damage costs. The change of the current simplifying approach that a flooding incident definitely fills the reservoirs upstream of any dam of any height in any location can be beneficial, as well. Moreover, the flood protection quantification could include other parameters regarding the flow, such as the flow velocity, temporary storage in the reservoirs, the peak or average flow-rate, sediment transport, etc. The aforementioned, though, require the use of simulation tools like HEC-RAS for the flow simulation to pre-calculate the flows for all the combinations of decision variables, which is impossible. This could be addressed in two ways: either empirically reduce the number of decision variables (exclude not probable dam locations, decrease available heights, pre-define the BL, etc.), or actually link the HEC-RAS software to the optimization tool and programmatically control it, triggering simulation of the proposed scheme (chromosome/solution) at each genetic algorithm iteration. This conceptual model/objective function upgrade will also unlock the refinement of the more realistic representation of artificial recharge benefit since the real volumes of upstream stored water to infiltrate will be calculated. The realism of the proposed upgrades requires the use of accurate predictions or historical data concerning hydrographs of flooding events in the studied area.

Another future challenge would be the use of the current methodology to optimize a small dams’ scheme, where the dams would be sediment retention dams and the maximization of sediment control would be an additional objective.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.