Maximizing Hydropower Generation in Flood Control Operation using Preference Based Multi-objective Evolutionary Algorithm

Reservoir flood control operation (RFCO) is a challenging optimization problem with multiple objectives which are conflicting with each other. In this work, a tri-objective optimization model which maximizes hydropower generation during flood control is investigated and a multi-objective optimizer based on decomposition technique and the decision maker’s preference information is developed. Experimental studies on typical floods at Ankang reservoir indicate the effectiveness of the proposed algorithm. It successfully provides preferred scheduling plans with more hydro- power generations than those considering the safety objectives only.

As a consequence of therapid progress in the last few years in multi-objective optimization techniques [3], more and more research efforts have been devoted to optimizing the conflicting optimization objectives in RFCO problems simultaneously, instead of converting it into single objective optimization problem as before. Yu et al. developed a multi-objective fuzzy decision-making model for RFCO problem [4]. Kim et al. proposed a multi-objective optimizer for RFCO problem using genetic algorithm [5].Nagesh-Kumar et al. presented a particle swarm optimization algorithm to optimize the reservoir operation policies [6].Afshara et al. suggested a multi-objective optimizer for RFCO problem using ant colony optimization algorithm [7]. Li et al. developed a shuffled frog leaping algorithm for multi-objective RFCO problem [8]. Qin et al. proposed a bi-objective model for RFCO problem and developed a cultured differential evolution algorithm to solve it [2]. Qi et al. developed an immune inspired algorithm for multi-objective RFCO problem [9]. Luo et al. suggested a hybrid multi-objective algorithm for RFCO problem by combing the particle swarm optimization and the estimation of distribution algorithm together [10].
Despite the success of multi-objective optimization algorithms for RFCO problem, almost all of them were developed with the aim of obtaining a set of trade-off solutions that approximates the entire Pareto front (PF) which is the set of all the best trade-off solutions. Few efforts have been devoted to incorporate the decision maker's (DM) preference into multi-objective optimizers for RFCO problems, although the preference based multi-objective optimization is not brand new in the community of multi-criteria decision making [11]. According to our previous investigations [9] [12], multi-objective RFCO  PFs which pose a big challenge to multi-objective optimization algorithms [13]. Moreover, in multi-objective RFCO problems, the difficulties of obtaining solutions at different PF regions could be various. In this case, multi-objective optimizers can hardly obtain a set of non-dominated solutions with good enough coverage on the PF. Instead, algorithms can converge to different PF regions on different flood instances without control. Such situation becomes even worse when solving a tri-objective optimization model for RFCO problem which considers hydropower generation in flood control.
Since it is challenging and unnecessary to approximate the entire PF in multi-objective RFCO problems, a preference based optimizer which obtains trade-off solutions at the preferred PF region could be a promising substitute. In our previous work, we considered the final upstream water level (FUWL) preference in bi-objective RFCO problem and developed a preference based selection mechanism for immune inspired multi-objective optimizer [14]. On the other hand, we developed preference based multi-objective optimization algorithm using decomposit-ion technique and applied it to solve bi-objective RFCO problem. In this algorithm, the DM's preference information is expressed by using the light beam search preference model [17] in the objective space. However, in RFCO problem, the FUWL preference is implicit, it is difficult to articulate in the objective spaceby using existing preference representation techniques. This paper extends our previous work and contributes in three aspects. 1) The objective of hydropower generation is considered in flood control and a tri-objective optimization model for RFCO problem is developed.2) A preference modeling method which converts the FUWL preference into a preferred region in the objective space is designed. 3) According to the preferred region which is dynamically determined by the proposed preference modeling method, a preference based multi-objective evolutionary algorithm is developed for RFCO problem.
The remainder of this paper is organized as follows. Section 2 describes the tri-objective optimization model for RFCO problem. Section 3 presents the preference modeling method for presenting the FUWL preference in RFCO problem. Section 4 describes the flowchart of the proposed algorithm. Section 5 verifies the effectiveness of the proposed algorithm. Section 6 conclusions this paper.

2.Multi-objective Model for RFCO Problem
Taking thehydropower generation and the FUWL preference into consideration, this work investigates the following tri-objective optimization model for RFCO problem.
Subject to: is the water release volumes at T scheduling periods. Each t Q (t=1, 2,…,T) has a non-negative value no larger than max Q . t Z is the upstream water level of the t-th scheduling period,it has a value between min Z and max Z . t V and t I are respectively the reservoir storages and the reservoir's inflow volume of the t-th scheduling period. FL Z isthe final target upstream water level at the end of the scheduling. E is the hydropower generation, it can be calculated as following.
where t ∆ is the time interval of the scheduling period. K is the efficiency coefficient. t Q and t H are respectively the water release volume and the water head of the t-th scheduling period. The average output t N has a lower bound min N and an upper bound max N .
In the optimization model in equation (1), 1 ( ) f Q means the highest upstream water level. It should be minimized to guarantee the safety of the upstream side. 2 ( ) f Q is the largest water release volume. We minimized it to protect the downstream side. 3

( )
f Q is the reciprocal of hydropower generation. We convert the maximization of hydropower generation during flood into minimization of its reciprocal.

Presentation of DM's Preference
The proposed preference modeling method converts the FUWL preference into a preferred region in the objective space. At each iteration of the multi-objective optimization algorithm for RFCO problem, the individuals whose final upstream water levels lie in between , , As shown inequation (3), the preferred region is defined based on the first two objectives, because the objective value of hydropower generation is the larger the better. In equation (4),

( )
FL Q denotes the final upstream water level of the individual Q. The middle point is defined as the objective vector of the individual whose final upstream water level is the closest to FL Z . The veto threshold vector extends the calculated resultV by 20%. The motivation behind this extension is to provide better coverage of the preferred region. Based on the definition of the preferred region, a measure of preference level, denoted asPL(Q), is defined on each individual Q to rank the preference degree of individuals.

4.The Proposed Algorithm
Due to its simplicity and efficiency, the multi-objective evolutionary algorithm based on decomposition (MOEA/D) has achieved a great success and attracted a lot of attention [17]. MOEA/D decomposes the target multi-objective optimization problem (MOP) intoa number of scalar optimization sub-problems by employing an aggregation approach with a set of evenly scattered weight vectors. And then, these decomposed sub-problems are solved simultaneously in a collaborative manner by using an evolutionary algorithm. We have also defined the WS-transformation which is a map from λ to ′ λ as following [13].
It should be noted that the WS-transformation is self-inverse (i.e. ). Based on this property, if we want to obtain some non-dominated solutions located within preferred PF region, we can generate some scalar optimization sub-problems with specific weight vectors which take those preferred solutions as optimal solutions. That is the basic idea of the preference based MOEA/D (p-MOEA/D) we developed in our previous work [15]. p-MOEA/D guides the search of MOEA/D towards the preferred PF region by removing scalar sub-problems from unwanted PF areas and add new ones into the preferred region.
This work inherits the preference based weight adjustment strategy in p-MOEA/D, design a new preference modeling method for representing the FUWL preference in tri-objective RFCO problem, and develop a preference and decomposition based multi-objective optimizer for RFCO problem (PD-RFCO). At each iteration, PD-RFCO maintains the following.  In algorithm1, the initialization step (line 1) and the evolving step (line 4) are exactly the same as those in MOEA/D [17]. In this work, the simulated binary crossover and the polynomial mutation are employed to generate new individuals. The ideal point is also updated in the evolving step.
In the UpdateEP step (line 5), non-dominated solutions are first identified from the union set of current EP and pop, giving rise to a temporary set of NS. If the size of NS is no larger than N E , then copy NS to EP. Otherwise, calculate the preferencelevel of each individual in NS using to equation (7), and update EP by the individuals with the first N E smallest preference levels in NS.
In the UpdatePreferrdRegion step (line 6), the preferred individuals with preference levels smaller than 1 are first identified from EP to form the population SP. If SP contains more than two individuals, then update the middle point M and the veto threshold vector V according to equations (4) and (5)(6) respectively.
In the RemoveSubproblems step (line 8), N WA scalar sub-problems are removed from the evolving population. It chooses the scalar sub-problems that will be removed according to their preference neighbor distances. The ones locate outside the preferred PF region will be selected in priority. However, when preferred solutions will have to be removed, the ones with smaller crowding distances will be selected in the first place.
In the AddNewSubproblems step (line 9), N WA individuals in the external population will be recalled into the evolving population and new weight vectors will be generated and added to the sub-problem set. On the contrary to the RemoveSubproblems step, the preferred solutions with smaller crowding distances will be selected in priority. When there are not enough preferred solutions, the ones with smaller preference neighbor distances will be selected in the first place.

5.Experimental Studies
In this section, typical floods at Ankang reservoir on the Hanjiang river in Shanxi Provence of China are investigated. The Ankang reservoir has amaximum water capacity of 2.585 billion cubic meters, a normal water level of 330 meters, a flood control limit level of 325 meters, a dead water level of 300 meters and a designed discharge capacity of 37474 cubic meters per second. The annual hydropower generation of Ankang station is about 2.857 billion KWH.
The parameters of the proposed PD-RFCO are set as follows. The evolving population size N is set to 100 and the external population size N E is set to 250. The neighborhood size T is set to 10. The initial middle point M 0 =(325,10000,-) and the veto threshold vectorV 0 =(4, 1000,-). The maximal number of adjusted scalar sub-problems N WA is set to 20. The iteration interval of weight adjustment I WA is set to 500. The maximal function evaluation number FE max is set to 10 6 times. The final target upstream water level at the end of the scheduling FL Z is set to 325. Thepreference threshold PT Z is set to 2.    PD-RFCO successfully converge to a preferred region on the estimated PF. When looking at the distribution of the obtained non-dominated solutions within the preferred region, it has a much simpler shape comparing with the entire PF. Since the PF shape has a significant influence on the difficulty of a MOP, it is very challenging for a multi-objective optimizer to approximate the entire PF of the tri-objective RFCO problem.
If we focus the search effort of the algorithm on a sub-region which is less complex in shape, the problem becomes easier. In addition, it is unnecessary to obtain non-dominated solutions that cover the entire PF, because only the ones within the preferred region are likely to be put into practice. Therefore, computing   Figure 3 is the discharge volumes of the non-dominated solutions obtained by PD-RFCO. It can be seen that the scheduling schemes have maximum discharging volumes of less than 6000 m 3 /s and 15000 m 3 /s respectively for the two investigated floods, which are much lower than the peak inflow volumes of the two floods, say 12200 m 3 /s and 21000 m 3 /s respectively. Therefore, the proposed PD-RFCO successfully provides scheduling schemes that reduce the flood peak significantly.    Experimental studies have been conducted to make a comparison between the tri-objective optimization model for RFCO in equation (1) and the existing bi-objective optimization model which does not consider hydropower generation and minimizes 1 ( ) f Q and 2 ( ) f Q in equation (1) only. Figure 5compares the hydropower generation of the non-dominated solutions obtained by PD-RFCO for solving the two optimization models respectively. In the box plots in this figure, the bottom and top of the box are respectively the first and third quartiles. The band inside the box is the second quartile, say the median. The ends of the whiskers represent the minimum and maximum of all the hydropower generation data. It can be seen that the boxes for the tri-objective model (right) are higher than those for the bi-objective model (left). This indicates the fact that by considering the maximization of hydropower generation in flood control, the optimizer can provide scheduling schemes with more hydropower generations.

Conclusions
In this work, a tri-objective optimization model which maximizes hydropower generation in flood control is developed for reservoir flood control operation (RFCO) problem. Considering the final upstream water level (FUWL) preference in RFCO problem, a new preference modeling method is designed by convertingthe FUWL preference into a preferred region in the objective space. The preference model is then incorporated into the algorithmic framework of the decomposition based multi-objective optimization algorithm (MOEA/D), giving rise to a preference and decomposition based multi-objective optimizer for RFCO problem (PD-RFCO).
Experimental studies have been done on two typical floods at Ankang reservoir to verify the effectiveness of the proposed PD-RFCO. Experimental results indicate that PD-RFCOobtains a set of non-dominated solutions which evenly scattered over the specified preferred region. And the scheduling schemes provided by PD-RFCO reduce the flood peak significantly. Comparison studies on multi-objective optimization models which consider and not consider the objective of hydropower generation have also been conducted.