Improved wind-driven optimization algorithm for the optimization of hydropower generation from a reservoir

A major crisis of the 21st century is the huge water demand caused by the large population growth (Srinivasan et al. 2012; Kummu et al. 2016). The energy demand for human society is also continuously increasing with the rapid development of social economy (Barra et al. 2013). As a renewable and clean source of energy, water resources are one of the critical tools for dealing with the energy crisis (Mehta et al. 2012; Tayebiyan et al. 2016). Reservoir operation optimization (ROO) is employed for the regulation and management of water resources, and it can lead to comprehensive economic benefits. It is also of great significance for meeting water and energy demands in the future (Tayebiyan et al. 2016; Nair & Sasikumar 2019).

For the ROO problem, a proper optimization algorithm and even just a small improvement in it can greatly increase the efficiency of power generation (Tayebiyan et al. 2016). Various optimization algorithms have been developed and applied to the ROO problem (Afshar et al. 2007; Bashiri-Atrabi et al. 2015; Asgari et al. 2016; Bozorg-Haddad et al. 2016). The optimal algorithms commonly used in ROO can be classified into the classic and evolutionary algorithms (EAs). The classic algorithms include linear programming (LP) (Lee et al. 2008; Feng et al. 2018), nonlinear programming (NLP) (Chu & Yeh 1978; Arunkumar & Jothiprakash 2012) and dynamic programming (DP) (Zhang et al. 2013, 2015). LP is the simplest and most commonly used method in ROO. However, the linear form requirement of LP does not suit the nonlinear characteristics of ROO. Thus, the LP optimization results are usually problematic. NLP can solve the nonlinear problems but it has the limitation that the objective function and constraints must be differentiable and some approximations must be made to get the solution. This can lead to local optimum and poor convergence performance of NLP. DP is not affected by function discontinuity, nonlinearity and problem randomness. However, the computational cost of DP is greatly increased with the increase in the reservoir group scale, which leads to the so-called curse of dimensionality problem (Morgenstern & Bellman 1962). Discrete differential DP, DP successive approximations and progressive optimization algorithm are some of the modified DP algorithms which are developed to reduce its high computational cost (Opan 2011; Lu et al. 2013; Zhang et al. 2015). However, these improved algorithms suffer from the shortcoming of the initial trajectory impaction on the final results (Feng et al. 2017; Changming et al. 2018). The EAs (Zitzler & Thiele 1998; Adeyemo & Stretch 2018), which benefit from the rapid development of computer technology, can satisfy the high-dimensional and nonlinear characteristics of ROO. Moreover, by using the constraint handling method, they can easily deal with the stringent constraint in ROO. EAs can obtain the Pareto-optimal solution, which means that within a reasonable calculation time there is no better solution in the search space and under all objectives. Due to the above advantages, EAs have been widely used in ROO in recent years. The most commonly used EAs are genetic algorithm (Chang et al. 2010; Akbari-Alashti et al. 2014), which simulates the natural gene and natural selection mechanism; cuckoo search algorithm (Ming et al. 2015; Meng et al. 2019), which simulates the brood parasitism of cuckoo; and particle swarm optimization (PSO) algorithm (Zhang et al. 2014; Niu et al. 2018), which simulates the foraging behavior of birds.

Whereas there is never the only best algorithm for a problem (Barra et al. 2013), and researchers have been trying to find a more efficient algorithm for ROO (Luo et al. 2014; Bashiri-Atrabi et al. 2015; Asgari et al. 2016; Bozorg-Haddad et al. 2017). In this paper, a novel algorithm, named wind-driven optimization (WDO), is introduced for ROO for the first time. Advantages such as clear concept, high efficiency, strong robustness and ease to implementation make WDO a widely used algorithm and some fields like scheduling residential loads of power grid (Javaid et al. 2017), segmentation and extraction of a carotid ultrasound image in medicine (Nagaraj et al. 2018) and the path planning of mobile robot (Pandey & Parhi 2017). Similar to other EAs, WDO still has the drawback of stagnating at local optima (Bao et al. 2015). To solve this issue, an improved WDO (IWDO) is proposed using the dynamic adaptive random mutation mechanism (RMM). Improving the algorithm efficiency and relieving the computation burden is another important problem when using EAs. The search space reduction (SSR) strategy of the decision variable through constraints transformation is usually employed to solve this problem (Li et al. 2015; Meng et al. 2019). However, the feasible region of SSR is computed before the start of EAs in these studies, which leads to a fixed feasible region for the strategies. In fact, the corresponding feasible space of the decision variable in each iteration is not always the same. An adaptive SSR (ASSR) strategy during the whole iteration process is introduced in this paper. ASSR narrows down the feasible region and improves the search efficiency. To summarize, the WDO algorithm is modified by a dynamic adaptive RMM, which reduces the chances of convergence to local optima and thus obtaining IWDO. Moreover, the ASSR strategy is used to improve the algorithm efficiency. The proposed algorithm and strategy for ROO are assessed by a case study.

The flowchart as shown in Figure 1 contains the reservoir operation model and the introduced algorithm for ROO. It represents the following items:

1. Reservoir operation model with the objective function and various constraints (Section 2.1).

2. WDO and IWDO algorithms (Section 2.2).

3. ASSR for ROO in the whole process (Section 2.3).

4. Specific procedure of WDO for ROO (Section 2.4).

The above items will be explained in the corresponding section in detail. Table 1 is the list of parameters and abbreviations that have been used in this paper.

1. Water mass balance equation constraint

   

   (2)
2. Water-level constraint

   

   (3)
3. Outflow constraint

   

   (4)
4. Generation output constraint

   

   (5)

   where (i.e. terminal storage volume of the t-th period) is decided by (initial storage volume of the t-th period), (average inflow of the t-th period) and ⁠.⁠, and are the lower limits of (water level of the t-th period), and (output of the t-th period), respectively. are the upper limits of ⁠, and⁠, respectively.

5. Nonzero constraint

All variables must be positive.

See Figure 2 to better understand the variables and the reservoir system. From Figure 2, we can see that when the outflow is greater than inflow, the water level, water storage and power generation head will decrease. If the outflow remains the same, as the decrease of power generation head, the power generation output will also decrease.

It is seen from Equation (6) that the velocity (⁠⁠) of new generation is determined by four terms. The first term represents the friction effect on air particles, and it reflects the inheritance of the current velocity. The second term represents the effect of gravity which is related to self-cognition of air particles and speeds up the convergence speed. Gravity enables the air particles to approach the coordinate center in proportion to the constant from the current position ⁠, which is a reflection of enhancing the learning process. The third term is the ‘detection’ of the air particle position. Under the influence of the pressure gradient force, the air particle moves to the optimal pressure point (i.e. the global optimal position) and considers its past experience comprehensively to make the next decision. is the Pareto-optima until generation P. j is the rank of the particles which is in ascending order by the pressure at a location. R is related to universal gas constant, and T is related to temperature. The fourth term provides the Coriolis effect on air particles, which enables the air particles to adjust their position properly in combination with the experience of their peers and, thus, to avoid the local optima. is the air particle velocity of any dimension other than t.

Equation (7) shows that the position (⁠⁠) of the new generation is equal to the current position plus the product of the new velocity and the unit time step ⁠. For simplicity, ⁠.

Users set the iteration number, population, ⁠, ⁠, and of the WDO algorithm. For details of the WDO algorithm, please refer to Bayraktar et al. (2013).

As stated earlier in Section 2.2.1, the air particles constantly move to reach an equilibrium state. This means that once an external force acts on the air particles that are in equilibrium, they will move again. Inspired by this, a disturbance, named dynamic adaptive RMM to local optima, is used to overcome the premature convergence and obtain IWDO.

In the next step, the position is updated according to Equation (7). The new position obtained after the random mutation is then compared with the previous optimal position. If the new position is better, it is used to replace the former optimal one. Otherwise, the previous optimal position is retained until the algorithm computation is over.

There are several constraints in various nonlinear optimization problems. While the so-called soft constraints can be violated, the violation of hard constraints results in invalid solutions (Chang et al. 2010). Due to the existence of the hard constraints in our problem, a large number of solutions are invalid, if they are not handled properly, the data redundancy will greatly reduce the algorithm's operational efficiency. To solve this issue, an ASSR strategy is employed in this study. For ROO, water storage at the end of each period is selected as the decision variable. We can see from Figure 1 that there is a one-to-one correspondence between the water storage and the water level; thus, the core idea of SSR is to obtain the feasible region of water level, which is to obtain the upper and lower limits of water level according to the water balance equation and considering various constraints in the solution process, and make sure that all the solutions generated during the iteration are within the feasible region (Barkat Ullah et al. 2008).

The approach for obtaining the feasible region is as follows:

1. Update the value: In each iteration, only the odd or even dimension water level is updated. In the P-th iteration, through updating the particle position at the end of period t − 1 and t + 1, and are generated (2 ≤ t ≤ N − 1).

2. Value inheritance: In generation P + 1, equals to and equals to ⁠, and only should be updated.

3. Determine the feasible region: Due to the fixed value of ⁠, ⁠, there must be a maximum and minimum value of based on Equation (2):

   

   (10)

   

   (11)

By using the method above, the violation of the constraints can be observed in the fitness function. A slight decrease will happen in the fitness function when there is a tiny degree of violation, while a notable decrease will happen when there is a high degree of violation. The mechanism for handling the constraints is represented in Figure 4.

Two conditions are explicated in the following. Condition 1: (1A) there is water surplus in the t-th period and not in the (t + 1)-th period, when water level at the end of the t-th period in the next iteration should rise but the first three terms are negative values; or, (1B) there is water surplus in the (t + 1)-th period and not in the t-th period, when water level at the end of the t-th period in the next iteration should decrease but the first three terms are positive values. Condition 2: any circumstances other than situation #1.

The fluctuation range and the value of water level are updated using Equations (22) and (23), respectively. The fitness value of is calculated to find the Pareto solution, and RMM is used to avoid the premature solutions. The algorithm then goes to the next generation until the global optimal solution is obtained.

Update water storage by the ASSR method as described in Section 2.3.

The IWDO algorithm is terminated after reaching the maximum number of iterations or termination conditions.

From the above introduction, the processing flow of the IWDO with the ASSR can be obtained. See Figure 5 for it. First, we set the parameters of the algorithm, and then the algorithm can initialize the individuals which is used in the first iteration. Then, the algorithm will calculate the fitness of each individual, and the one with the maximum fitness value will be chosen as the optimal one. Later, the algorithm will judge whether it is premature. If it is premature, RMM will be used to update the velocity and position of the local optima, and then adjust the new optima to the feasible region which is obtained through ASSR. If it is not premature, use the ASSR strategy to update the new generation. Up to now, the new generation is updated and the algorithm will terminate and output the optimal solution if it has reached the terminal condition. If the algorithm does not reach the terminal condition, it will be back to the new cycle.

The Wujiang River, with a total length of 1,037 km and a drainage area of 87,920 km², is the largest tributary on the South Bank of the upper reaches of the Yangtze River. The Wujiang River originates from the Yunnan-Guizhou Plateau, flows through Hubei and Chongqing and finally joins the Yangtze River. The altitude of the basin is high in the southwest and low in the northeast, with rough topography and deep valley. Most of the area of the Wujiang River Basin belongs to the subtropical monsoon climate, with an average annual rainfall of 1,163 mm (Wenhui et al. 2015; Xiao et al. 2015).

The Goupitan hydropower station (GHS) is located in the middle reaches of the Wujiang River. Figure 6 shows the location of GHS. The purpose of GHS is mainly for power generation, followed by shipping and flood control. The GHS has an annual regulation capacity. It is the largest hydropower station in the Guizhou Province and the whole mainstream of the Wujiang River. The drainage area is 43,250 km², the total storage capacity is 6.454 billion m³, and the total installed capacity is 3,000 MW; the firm output is 600 MW and the minimum outflow is 300 m³/s. The operation period is from June to May of the following year, during which the flood period is from June to October. Twelve time intervals are set in months in a scheduling period for the operation. During the flood season, the flood control limit water level in June and July is 626 and 628 m in August. In other months, the upper limit of water level is 630 m. After the flood season, the incoming water gradually decreases, and thus, the water level should be kept at a high level to improve the power generation. However, in addition to the hydraulic head, the power generation is also related to the outflow. Therefore, ROO is employed to produce optimal results. The key parameters of GHS are given in Table 2.

In our case study, an optimal solution is obtained when parameters of the IWDO algorithm are chosen as α = 0.05, g = 0.5, RT = 0.1, maximum iteration number = 500 and population size = 100. The performance of the WDO and IWDO algorithms with PSO is compared. When the parameters of PSO algorithm are = 0.729, C₁ = 2 and C₁ = 2, it can obtain the best solution according to Mendes et al. (2004). The population size and iteration number of all algorithms are the same. It is noted that the same fitness function is used for all algorithms. The used parameters for each algorithm are in Table 3.

Apply ASSR to WDO, PSO and IWDO to get ASSR-WDO, ASSR-PSO and ASSR-IWDO, respectively. Each algorithm for ROO is computed 10 times, and the standard deviation (SD) of power generation is used to analyze the robustness of algorithms. The best value, average value and SD of power generation are shown in Table 4, from which it can be found that the SDs of WDO and PSO are zero, indicating that the standard algorithms are in good robustness; the SD of IWDO is 0.01 and the SD of ASSR-IWDO is 0.001, indicating that RMM will cause some deviation but ASSR can reduce that.

The efficiency of WDO and IWDO for the ROO problem can be assessed using the best scheduling results reported in Table 5. Results of PSO are also reported in this table as a reference for comparison. It can be seen that the results computed by all algorithms meet the hard constraints, indicating that they are all effective for ROO. Moreover, the total power generation obtained by IWDO is higher than WDO, which demonstrates the improvement caused by RMM. Moreover, it can be seen that the total power generation of WDO is higher than PSO. It is noted that the outflow and output of WDO in March are below the minimum value. See also February and March in the case of IWDO, and March and April in the case of PSO. Further research is required to overcome such shortcomings of the algorithms.

The optimal scheduling results of ASSR-WDO, ASSR-IWDO and ASSR-PSO are given in Table 6. It can be seen that all three algorithms combined with ASSR can fully satisfy all the constraints in every period, meaning that ASSR can ensure that the scheduling results of each algorithm are reasonable. As such, ASSR improves the solution quality. Figures 7–9 show the change of water level, discharge and output, respectively, and from which we can see that the water level in each period is higher than the dead level and lower than the max level, and the discharge and the output in each period are greater than the minimum value. The total power generation obtained by ASSR-WDO, ASSR-IWDO and ASSR-PSO is 9.88, 9.92 and 9.87 billion kw·h, respectively, proving that ASSR-IWDO can get the highest generation.

Figure 10 shows the convergence curve of the fitness value of all three algorithms. It can be seen that for ASSR-IWDO and IWDO, the initial fitness value is the same, but the final fitness value of ASSR-IWDO is much higher than that of IWDO. The number of iterations to reach the stable convergence in ASSR-IWDO is about 50% of that of IWDO, the same regularity of ASSR-WDO and WDO, which demonstrates that ASSR can improve the search efficiency. Moreover, a comparison of ASSR-PSO and PSO shows that ASSR can be successfully adopted for other algorithms.

In this study, WDO is first employed to ROO. An IWDO algorithm is further developed by using a dynamic adaptive RMM to avoid local optima. Furthermore, an ASSR strategy is proposed to improve the search efficiency of the algorithm. To generate a compressed feasible search region, ASSR only updates the value of odd or even dimension in each iteration, and it combines the water storage inherited from the previous iteration with the minimum outflow, and then making the use of the water balance equation to obtain the upper and lower limit of the water storage which is to be updated in the current iteration. Moreover, the efficiency of WDO and IWDO are evaluated in comparison to the PSO algorithm. The ASSR is also applied to the PSO algorithm to assess the performance of ASSR. Combined with ASSR, three algorithms are generated, including ASSR-WDO, ASSR-IWDO and ASSR-PSO. Using the GHS as the case study to evaluate the performance of the proposed algorithm, the following conclusions can be drawn:

1. IWDO is an effective algorithm for ROO. All three algorithms of WDO, IWDO and PSO meet the hard constraints. However, it is noted that in some months, the outflow and the output are below the lower limit.

2. In terms of efficiency, IWDO is superior to WDO, and WDO is superior to PSO. Comparing the obtained power generation shows that IWDO is capable of obtaining better solutions than WDO, and so is WDO than PSO.

3. ASSR can improve the search efficiency and search quality. The results of all algorithms combined with the ASSR strategy meet all the constraints. Moreover, ASSR-IWDO shows faster convergence and greater power generation compared to IWDO. The same result is presented when comparing ASSR-PSO with PSO. Thus, it is concluded that ASSR can be integrated into any optimization problem to enhance the search efficiency. The performance of ASSR-IWDO in solving the cascade ROO problem will be evaluated in a future study.

This study is supported by the National Key Research and Development Program of China (No. 2018YFC1508004). We highly thank the editor and anonymous reviewers for their constructive comments.

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