Algorithm and architecture of the hybrid model

c) Hybrid Algorithm (GA-NLP)

The Hybrid Algorithm, (GA-NLP), performs the design of a branched CIPN n operating with the shift mode, simultaneously optimizing the shifts assigned to each hydrant (using GA) and the dimensioning of the pipes of the network (through NLP). The objective is to minimize the system’s cost, exploiting in this way the best characteristics of each type of algorithm. Figure 1 shows the flow diagram of the GA-NLP method.

Figure 1 Flow chart of the GA-NLP optimization method. 

Through GA the shift belonging to each hydrant is found that minimizes the objective function, which is the cost of pipes in the network. In this way, the decision variables are exclusively the allocation of shifts to hydrants, and the dimensionality of the problem in the GA formulation is reduced. In each step of the GA, the tentative turns assigned to hydrants are evolved, using the cost of the network for the allocation of candidate shifts as an assessment of the aptitude of each individual. This process was implemented in the EVOLVER package (^{Palisade Corporation, 2010}), which allows the incorporation of optimization processes through GA to conveniently codified problems of a general nature.

In turn, the cost of the network for an allocation of candidate shifts is the result of pipeline optimization using the GRG analytical algorithm with NLP. The function of the GRG algorithm is to calculate the pipeline distribution of continuous diameters in the network, which satisfies the pressure requirements and generates a minimum cost for a certain allocation of shifts to the hydrants. The GRG is incorporated in the SOLVER package (^{Frontline Systems Inc., 2011}), a software package for general purpose simulation and optimization. In this work, a linkage of both algorithms has been made that allowed for the communication of results obtained by SOLVER for the network as values of the objective function used later by EVOLVER for the implementation of the GA, as shown in Figure 1, where P ₀ represents the initial population and P _t the resulting population for the process in the GA.

Once the final process solution is determined, with the diameters in each line, proceed to:

• Normalize the theoretical diameters by the criterion of the nearest diameter

• Check the resulting pressure values in each shift in all hydrants in the network with pressure restriction and readjust the diameter if necessary.

• Check the flow velocity in all the network pipes and readjust if necessary.

• Compute the cost of each pipe in the network according to the final standard diameter assigned.

Flexibility indicator of the CIPN to shifts

In the construction of the GA-NLP Hybrid Algorithm for the design of a CIPN, two criteria have been applied: (i) comply with the established pressure and (ii) minimize network costs with decision variables of both pipe diameter and the shift assignments. In the formulation of the objective function, no criteria of reliability or flexibility are introduced. However, in all shift networks, once they have been dimensioned, it is advisable to evaluate their capacity to continue providing adequate service in the event of shift change, a characteristic that will constitute their flexibility concerning shift changes. These changes frequently occur in the networks throughout their operation, and lead to the adoption of different combinations of hydrants assigned to a certain shift with respect to the initial assignment assumed in the design, respecting at least the total flow corresponding to the shift.

In ^{Lamaddalena and Sagardoy (2000)}, for CIPN on demand, there is an antecedent that defines indicators that assess whether the level of pressure contributed to the hydrants is sufficient according to the number of open hydrants, with total flows below or above the design conditions. It is possible to extend these concepts to CIPN designed with shift operation to evaluate the flexibility of the same to fit changes of shifts.

The shift change is an unpredictable event with a large space of possibilities. Therefore, the preparation of indicators should explore scenarios that contemplate any shift reorganization with the condition that the total flow demanded in the shift is maintained, corresponding to a certain global percentage of open hydrants. Provided that the network has a sufficient number of hydrants with similar endowments, a change in shift configuration can be assimilated to a random opening of a percentage of hydrants equal to that corresponding to each initial shift.

For each shift assignment (random or deterministic), the pressure sufficiency condition in each hydrant (n) during a shift (t) means that the following relationship is satisfied (^{Lamaddalena & Sagardoy, 2000}):

Where Hn,t is the hydrant pressure n within the shift t to which it belongs, expressed in meters, and Hconsn is the minimum required pressure or setpoint height of the hydrant n in meters.

The pressure compliance variable (CPn,t) in the hydrant n during the shift t it is defined as a binary value between 0 and 1, where the value 1 indicates that the pressure satisfaction condition is verified and the value 0 indicates that sufficient pressure does not reach. If a number of different scenarios N _E are generated that are associated with turn t in which the hydrant n participates, and CPn,ti is evaluated for each i scenario in each hydrant, the pressure reliability parameter in each hydrant (FPn) is defined as follows:

FPn represents the ratio of the number of scenarios or shifts in which the pressure was sufficient with respect to the total of possible shifts tested in which hydrant n participates. This parameter calculated for each hydrant identifies the hydrant’s vulnerability before changes in shift organization, and allows for discrimination if there are failures, sporadic or frequent, in only certain hydrants, or in all of them.

If there are a total of Nh hydrants in the network, the system flexibility indicator against shift change, IFCT, is defined as the average of the pressure reliability in all the hydrants.

The IFCT is a synthetic indicator that cushions possible pressure shortfalls in shift changes located in a few hydrants. If its value is high, (close to 1), it follows that all hydrants in the system maintain sufficient pressure before any shift change, indicating high CIPN flexibility. If the index is low (close to 0), the inability to supply sufficient pressure outside the design turn is frequent and affects many hydrants in the CIPN (low flexibility).

Sensitivity analysis for GA parameters

For this study, values of between 50 and 200 individuals for the set of selected networks have been taken as population sizes. Also, a crossing probability (P _c ) of 0.5 (^{Mora, 2012}) has been set as a reference value for the crossing operator. On the other hand, ^{Mora-Melia et al. (2013)} proposed to determine the probability of mutation (P _m ) based on the number of the problem’s decision variables. These authors proposed to take the inverse, that is, 1/N _h and define the space of feasible solutions to reduce the search time. Values close to 0.01 indicate that the process uses a large space to generate the optimal solution, while values close to 1 limit the search. The number of hydrants is different in each of the study networks, and this criterion will be used as a reference value. Thus, to adjust the GA, population values of 50, 100 and 200 individuals have been tested; crossing probabilities of 0.1 and 0.5, and P _m values between 1/N _VD and 0.9.

Comparison of results with alternative algorithms

To evaluate the effectiveness of the Hybrid Algorithm (GA-NLP), in terms of the cost of the designs produced, the total costs obtained in the optimal design process of a set of networks were compared using GA-NLP as well as three other alternative design methods.

The three alternative methods used variants of an algorithm based on the use of Lagrange Multipliers, (known in network design literature as Economic Series, MSEM). This method was improved for CIPN on demand (^{González & Aliod, 2003}) and later extended for CIPN to shifts (^{García et al., 2011}). This algorithm is implemented in the GESTAR 2016 software. GESTAR 2016 is an integrated package specifically for CIPN design and analysis. It is widely used in the projection and management of CIPN and used in literature to compare the costs of shift and demand-based designs and (^{Alduán & Monserrat, 2009}; ^{Monserrat et al., 2012}). One of the modules of this GESTAR application was also used to facilitate the computation of FP _n indices in the hydrants needed for IFCT calculation.

Since the GA-NLP hybrid method obtains continuous diameters which must be normalized at the end of the process, both results are incorporated into this work. The nomenclature and description of the compared design alternatives are listed below:

Method 1 (GA-NLP, continuous diameters): Assignment of optimal shifts through GA and network design using NLP, without diameter adjustment.

Method 2 (GA-NLP, normalized diameters): Method 1 plus diameter normalization, by automatic assignment to closer diameters and revision for the verification of minimum pressures and maximum speeds.

Method 3 (MSEM): Optimal shift sizing, through MSEM integrated in the GESTAR tool, with shift allocation obtained the Methods 1 and 2.

Method 4 (MSEM-tur-aleat): Optimal shift sizing, through MSEM integrated in the GESTAR tool, with random manual shift assignments (without optimization).

Method 5: Network design operating on-demand through MSEM integrated in the GESTAR tool, with total irrigation time equal to the sum of shift times.

Case studies

For the present study, four irrigation networks were selected, two located in Ecuador (San Rafael and Tuncarta) and 2 in Spain (Lower Callén and Valcuerna). Table 1 shows the number of hydrants, the irrigation surface, head reservoir surface elevation, setpoint pressures, continuous assumed flow rate, and the effective watering day of the four selected networks.

Figure 2 and Figure 3 show the topology of the different study networks. In these figures, the header node corresponding to the reservoir (node 0), the junction nodes in the network mainline, the demand nodes (hydrants), and the pipelines have been represented.

Figure 2 Topology irrigation systems: a. “San Rafael” b. “Tuncarta”, c. “Callén Inferior”. 

Figure 3 Topology of "Valcuerna" irrigation system. 

A summary of the characteristics of design shifts for 4 CIPN under study is presented in Table 2, which includes the time of shift duration and the head flow for each shift.

Sensitivity analysis and parameter calibration of the GA-NLP Hybrid Algorithm

The results of the design costs obtained by the Hybrid Algorithm GA-NLP for each of the Case Studies considered is shown in Table 3, based on two sets of GA configuration parameters according to what is stated in Section “Sensitivity analysis for GA parameters”. In the first set of parameters, called Uniform Parameters, the parameter configuration of the crossover probability (P _c ), mutation probability (P _m ) and population size is initially assigned by default and uniformly for all cases, with values of 0.1, 0.9 and 200, respectively. The second set, denoted as Calibrated Parameters corresponds to a combination of parameters calibrated so that, in each case, it leads to the minimum cost, also reducing calculation times, both configurations being found in the extreme ranges of P _m . The computer used has a Core i7 processor at 3.40 GHz with a RAM of 16 Gb. Table 3 includes the execution time of the algorithm, number of tests or executions, the total network design cost directly obtained by the GA-NLP with continuous pipe diameters (Method 1) and the total design cost once the diameters are normalized according to the procedure described in section “Comparison of results with alternative algorithms” (Method 2).

As can be seen, the differences in design cost with normalized diameters for the Case Studies, using values of uniform initial crossover and mutation operators, or calibrated parameters for each case, which suppose even extreme values, is small. However, the calculation time is significantly reduced when the parameters are calibrated, due to the associated population size (Table 3). Based on those mentioned above, for the following analyses and comparisons, the results obtained shall be adopted with uniform parameters, not calibrated "ad hoc" for each case, since they represent the most unfavorable condition. These results demonstrate the robustness and consistency of the GA-NLP Hybrid Algorithm presented to find viable solutions when applied to Case Studies of diversified characteristics even with very different P_c and P_m parameters, without needing to force the "ad hoc" introduction of perturbations in the mutation process as in ^{Farmani et al. (2007)} by the high dimensionality of the decision variables involved in the same GA both for the allocation of shifts and diameters.

Comparison of costs and flexibility of the GA-NLP Hybrid Algorithm designs and alternative design methods

The results, in terms of network pipe cost of the optimal designs of the Case Studies for all the Design Methods described in Section 3.2 are presented in Table 4. The IFCT value of the designed networks is also included with each one of said methods.

It is observed in Table 4, that even without considering Method 1, a direct result of applying the GA-NLP before the normalization of the diameters, said algorithm with subsequent normalization (Method 2) always obtains a lower cost than the design procedure for shifts optimized by traditional Economic Series-type analytical algorithms, with shifts assigned in a heuristic way (Method 4). When comparing the results of the cost of the networks by means of Method 2, with Method 3, in which shift allocation has been introduced by means of GA-NLP (Method 1 and 2) are similar values. This suggests the decisive influence of shifts allocation in cost reduction in networks designed to operate shifts, apparently this GA-NLP capacity being more relevant than the algorithm itself used in the dimensioning of the pipelines.

Once the effectiveness of the GA-NLP Hybrid Algorithm has been proven to generate cheaper shift-network designs than those found by traditional alternative methods, the flexibility achieved can be analyzed with the GA-NLP (Method 2) concerning the alternative design methods presented in this study. Table 5 shows the percentage variations of pipeline costs and the IFCT of Methods 3, 4 and 5 (M3, M4, M5 respectively in said Table) with respect to Method 2, which is taken as reference in all the variants. The percentages of positive cost increases (Δ Cost %) and negative IFTC increments (Δ IFCT %) imply higher price and less flexibility, respectively, of the alternative indicated with respect to the GA-NLP results (M2).

In Figure 4 (a, b), the percentage increases of cost (Δ Cost %) are correlated with the respective IFCT increase (Δ IFCT%) for Methods 3 and 4.

Figure 4 a) Relative cost variations vs. relative IFTC variations M3/M2 for the Case Studies; b) Relative cost variations vs. relative IFTC variations M4/M2 for the case studies. 

From the comparison between Method 3 (which has required information from the optimal shifts obtained from Method 2) and Method 2, it is observed in Table 5 (first column) and Figure 4a, that in the CIPN shift designs with Method 3, the costs are similar to Method 2 with more flexible designs. With Method 3, the network flexibility decreases by a total average of -5.37%, except the Lower Callén network, where flexibility increased slightly (0.92%).

Method 4 not only generates more expensive designs, but is also less flexible than Method 2, (see Table 5, column 2, and Figure 4b). There is an increasing trend in total network costs (average: 8.6%) and a reduction in flexibility (average: -7.50%). The reason is the lack of shift optimization involving Method 4, which again highlights the great importance of the shift allocation in the design concept.

All of this indicates that the GA-NLP Algorithm generates designs with not only cheaper results than the alternative design procedures for shuffled shifts, but also more flexibility in the face of overtime shift changes.

Shift design vs. demand design

As a collateral result, in the light of the previous analyses, a criterion has been inferred to help with rational decision-making when establishing the convenience of a shift-based or demand-based design of a new CIPN. This criterion consists of relating the design costs and flexibility to shifts (Method 2) with that of design to demand (Method 5).

With the data in Table 5, Column 3, Figure 5 is drawn. The abscissa axis represents the percentage of the IFCT increase from design to demand (Method 5) with respect to the shift design (Method 2). The ordinate axis indicates the cost-percentage increase of the design to the demand (Method 5) with respect to the design in shifts (Method 2). The diagonal line indicates the region where the increase in costs increases in a similar way to the increase in flexibility.

Figure 5 Relative cost variations vs. relative IFCT variations when moving to a demand design (M5) from a shift design (M2) for the case studies. 

In cases where the cost increases versus the flexibility increases (IFCT) are located above-said line, the shift-based design will be the most appropriate, since the design’s cost increases to the demand with respect to the shift design exceed the corresponding flexibility increases. The cases that located next to or below said line (shaded area of Figure 5) would point to a preference of demand design. These conditions can be formalized stating that the demand design would be recommended if:

Otherwise, it will be more advisable to form a shift design. This criterion cannot be taken in an absolute but in an indicative way, and it must be complemented with other important parameters in the decision-making, as presented in ^{Espinosa et al. (2016)}.

When applying this criterion to the 4 case studies, it is evident that in the Tuncarta and San Rafael networks the cost increases remarkably (average: 36%) without improving the flexibility, and in Figure 5 they are clearly above the shaded area. For these networks it is advisable to carry out the design with the shift modality. For the Lower Callén network, both the IFCP and the cost increased 20% and 30% respectively, reaching the threshold of the convenience of the demand design. Therefore, in such networks it might be advisable to design a network operating on demand to obtain better advantages in operating the system, considering the available resources. For the Valcuerna case, demand design cost less than the shift-based design. This is because hydrant opening times are very different and, consequently, demand-design flows are lower than those in shift-design. For this reason, the best option is the demand-design, unless both the number of shifts and their duration are redefined, and the provisions to homogenize the opening times of the hydrants are be modified.

There is a high variability of cost differences between demand and shift designs according to each case study’s network, with savings in shift design versus demand design located between 19% and 32%. These variations are in line with what has been observed in other investigations focused on establishing comparisons between demand and shift designs. In the network studied in ^{Espinosa et al. (2016)}, the savings in the shift system versus the demand system is 22%. In the network analyzed by ^{Alduán and Monserrat (2009)}, this saving is 15%, and in the three cases found in ^{Monserrat et al. (2012)} is in a range of 5% to 9%. The fact that this research has found generally greater savings is interpreted to be due to the fact that we have optimized the allocation of shifts, as is also the case presented in ^{Farmani et al. (2007)} in which savings amounted to 55%.