Multi-objective optimal operation of multi-reservoir systems is complex and multi-dimensional. This type of optimization requires reformulation (or extension) of the parent evolutionary algorithms to cope with specialized conditions and solve efficiently the targeted multi-objective, multi-reservoir, operation problems. The authors propose an extended MOFA, namely MODFA, that is particularly suitable for multi-objective, multi-reservoir, operation problems. First, the steps of the extended FA (namely, DAF as proposed by Garousi-Nejad et al. 2016b) are summarized, followed by the description of MODFA applying the NSGA-II approach.
• Step 1: The second and the third terms on the right-hand side of Equation (16) play the roles of modification of position and of random movement, respectively. Yang (2009) suggested that the values of and be set equal to 1 and in the range , respectively. However, Garousi-Nejad et al. (2016b) proposed that these two parameters introduce a trade-off. In other words, the proper values of and are such that they cause the value of in Equation (14) to vary between 0 and 1. Suitable values for and are found in Table 1 of the work by Garousi-Nejad et al. (2016b).

• Step 2: This step increases the conditions under which solutions are modified. It is seen in the flowchart of the MOFA shown in Figure 2 that if firefly i is not worse than firefly j in terms of attractiveness (i.e., objective function value), no modification is effected on firefly 's position. The MODFA, however, modifies firefly 's position through a random walk even if firefly i is not worse than firefly j in terms of attractiveness. This step produces more new positions of fireflies in each iteration.

• Step 3: MOFA implements a random walk by means of the third term on the right-hand side of Equation (16) in which a value in the range is applied on all decision variables of a firefly. This type of random walk is not suitable in reservoir operation problems because the decision variables are mostly large values. This calls for random walks that are larger than the range . Thus, contrary to MOFA in which the same random walk is effected on all decision variables, the MODFA first categorizes the variables randomly into groups (specified by the user) and each group is assigned a separate range. More details can be found in Garousi-Nejad et al. (2016b). Consequently, small and large random values are produced. It is noted that in the MODFA large values are damped and reduced during the execution of the algorithm. This modification in algorithmic Step 3 creates more solutions in the decision space in the first iterations.

• Step 4: According to Yang (2013), defined in Equation (15) is not limited to the Euclidean distance. In fact, any mathematical statement that can effectively characterize the quantities of interests in the optimization problems can be used as the distance depending on the type of the problem of interest. The term becomes nearly zero due to the large values of decision variables in reservoir problems (regardless of the value of ), and it causes the effect of the modification term to vanish. Therefore, Equation (17) is defined instead as the distance between fireflies in the MODFA:
17
in which and the values of objective functions of firefly i and firefly j, respectively.
• Step 5: The MODFA selects the last sorted solutions of each iteration that are transferred as the best solutions to the next iteration. In other words, solutions that are the same in terms of the decision variables' values (and not in terms of the same objective functions' values) are chosen once. Unlike the MOFA, this prevents quick attenuation.

• Step 6: There are also specific recommendations for selecting MODFA parameters in solving reservoir operation problems. The value of in MODFA is the same as recommended by Yang (2009) in the MOFA. The MODFA uses a range of equal to [1, 5]. The reason for choosing this range is that these values do not cause the value of to be greater than 1. Therefore, according to algorithmic Step 1, the modification term is not combined with the random walk. Garousi-Nejad et al. (2016b) provide a detail analysis of the proper values of . The complete flowchart of the MODFA is depicted in Figure 3, in which the reformulated (or extended) steps are shown with light gray rectangular shapes and other steps are the same as those of the MOFA.

Figure 3

The flowchart of the MODFA.

Figure 3

The flowchart of the MODFA.

Table 1

Data for three reservoirs and their power plants

CharacteristicsReservoir
Karoun 4Khersan 1Karoun 3
Purpose of dam Hydropower generation Hydropower generation Hydropower generation
Maximum reservoir level (masl) 1,025 1,013 840
Minimum reservoir level (masl) 990 1,000 800
Maximum reservoir volume (106 m32,019 332.55 2,252.58
Minimum reservoir volume (106 m31,144.29 262.68 1,110.12
Active reservoir volume (106 m3748.71 69.87 1,142.66
Maximum reservoir release (106 m3450 400 1,000
Power plant capacity (106 W) 1,000 584 2,000
Performance coefficient (%) 20 25 25
Power plant efficiency (%) 88 93 92
CharacteristicsReservoir
Karoun 4Khersan 1Karoun 3
Purpose of dam Hydropower generation Hydropower generation Hydropower generation
Maximum reservoir level (masl) 1,025 1,013 840
Minimum reservoir level (masl) 990 1,000 800
Maximum reservoir volume (106 m32,019 332.55 2,252.58
Minimum reservoir volume (106 m31,144.29 262.68 1,110.12
Active reservoir volume (106 m3748.71 69.87 1,142.66
Maximum reservoir release (106 m3450 400 1,000
Power plant capacity (106 W) 1,000 584 2,000
Performance coefficient (%) 20 25 25
Power plant efficiency (%) 88 93 92
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