# The Improved Binary-Real Coded Shuffled Frog Leaping Algorithm for Solving Short-Term Hydropower Generation Scheduling Problem in Large Hydropower Station

Authors: Zhe Yang; Kan Yang; Lyuwen Su; Hu Hu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The short-term hydro generation scheduling (STHGS) decomposed into unit commitment (UC) and economic load dispatch (ELD) subproblems is complicated problem with integer optimization, which has characteristics of high dimension, nonlinear and complex hydraulic and electrical constraints. In this study, the improved binary-real coded shuffled frog leaping algorithm (IBR-SFLA) is proposed to effectively solve UC and ELD subproblems, respectively. For IB-SFLA, the new grouping strategy is applied to overcome the grouping shortage of SFLA, and modified search strategies for each type of frog subpopulation based on normal cloud model (NCM) and chaotic theory are introduced to enhance search performance. The initialization strategy with chaos theory and adaptive frog activation mechanism are presented to strengthen performance of IR-SFLA on ELD subproblem. Furthermore, to solve ELD subproblem, the optimal economic operation table is formed using IR-SFLA and invoked from database. Moreover, reserve capacity supplement and repair, and minimum on and off time repairing strategies are applied to handle complex constraints in STHGS. Finally, the coupled external and internal model corresponding to UC and ELD subproblems is established and applied to solve STHGS problem in Three Gorges hydropower station. Simulation results obtained from IBR-SFLA are better than other compared algorithms with less water consumption. In conclusion, to solve STHGS optimization problem, the proposed IBR-SFLA presents outstanding performance on solution precision and convergence speed compared to traditional SFLA effectively and outperforms the rivals to get higher precision solution with improving the utilization rate of waterpower resources.


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## Body

## 1. Introduction

Short-term hydro generation scheduling (STHGS), i.e., economic operation in hydropower station, including space and time optimization, is a typical dual optimization, multidimensional, noncontinuous, nonlinear problem with numerous complex hydraulic and electrical constraints. STHGS aims to find a profitable way to decrease water consumption and the cost of on and off transition between hydropower units with satisfying the power generation demand. The STHGS problem can be decomposed into two subproblems: the space optimization, namely, economic load dispatch (ELD), and the time optimization, the hydropower unit commitment (UC). The ELD subproblem attempts to reasonably determine the power output of each unit so as to minimize water consumption [1]. Efficiency, under different operation conditions, corresponding to each type of units is discrepant especially the large capacity unit. In addition, to keep unit operational within stable operation zone is beneficial to prolong the life of units. Therefore, to effectively distribute output for a startup unit to burden and focus energy on improving integrated units operation efficiency in stable operating zone are of vital importance in STHGS problem. The unit commitment (UC), aiming to pick up a wise startup and shutdown schedule of units during operational period, is 0-1 binary coded subproblem [2]. Along with the power generation, a large amount of water will be consumed in water turbine. Moreover, part of the water will be lost due to state change between startup and shutdown and frequent on and off transition is greatly harmful to service life of units [3]. Thus, a rational and highly efficient unit commitment can steeply minimize water consumption and simultaneously improve dependability of safe operation in hydropower system. In summary, to accurately describe and find reasonable unit commitment and economic load distribution can not only save the water resources and operation cost but also improve the turbine efficiency and service life. In order to efficiently utilize hydropower resources in reservoir and save limited water resources, it is meaningful and significant to build the model of STHGS and acquire feasible solution with effective method.During the past few years, a number of traditionally mathematical methods have been put forward to handle STHGS problem. The traditional methods are focused on branch and bound algorithm [4] (Cohen and Yoshimura 1987), linear programming (LP) [5] (Jabr et al. 2000), mixed integer nonlinear programming [6] (Catalao J 2011), Lagrangian relaxation (LR) [7–10] (Oliveira et al. 1992; Ohishi T, W. Ongsakul; N. Petcharaks; Sum-Im et al. 2014; Cheng-C-P), quadratic programming (QP) [11] (Granelli and Montagna 2000), and dynamic programming (DP) [12–15] (Allen and Bridgeman 1986; Siu et al. 2001; Yi et al. 2003; Cheng et al. 2009), which have met with different degrees of success to STHGS. Nevertheless, methods mentioned have defects on mechanism, stability, and efficiency. Lagrangian multiplier is introduced into the objective function to structure dual problem to simplify calculation in the LR method; however, it is still difficult for coupling various constraints and finding a suitable multiplier. DP, though it can obtain the best solution theoretically, tends to face dimension disaster and costs plenty of computation time, which limits the application in dealing with STHGS issue. Therefore, the approximation of dynamic programming (ADP) algorithm is put forward to solve the problem of dimension disaster and overcomes these shortcomings to an extent [16] (Wei et al. 2014). On the whole, traditional methods encountered numerous obstacles when facing large scale power systems with huge installed capacity.To overcome shortage of traditional methods, along with high-speed computer technology development, intelligent algorithms, i.e., heuristic stochastic methods, have been developed rapidly to exploit application in managing STHGS problem. These methods mainly include evolutionary programming (EP) [17] (Christober et al. 2011;), differential evolution (DE) [18, 19] (Lu et al. 2010; Mo et al. 2013), genetic algorithm (GA) [1, 20, 21] (Abido et al. 2003; Senthil Kumar V and Mohan M.R 2010; Zheng et al. 2013), tabu search (TS) [22] (Bakhta N et al. 2013), artificial neural network [23] (Sasaki H et al. 1992); particle swarm optimization (PSO) [24–26] (Yuan et al. 2009; Kumar et al. 2011; Zhao (2006)), chaotic optimization algorithm (COA) [27] (Cai et al. 2007), immune algorithm [28] (Wang and Yang 2010), ant colony optimization (ACO) [29, 30] (Vaisakh K and Srinivas L (2011); Chen et al. 2015), gravitational search algorithm (GSA) [31–34] ( Shen et al. (2015, 2016 a, b); Yuan X. et al. 2014; Li et al. 2015), and fireworks algorithm (FA) [35](K. Srikanth et al. 2016). Owing to flexible search strategies and powerful convergence capability, these algorithms can obtain higher quality solution with accepted optimization time to a certain extent in comparison with traditional methods. Furthermore, the mixed algorithm which combines superiority of several algorithms is introduced to accelerate computational speed and boost search ability for global optimum. However, intelligent heuristic stochastic methods also cause defects and imperfect constituent parts. The improved immune algorithm is applied to optimize the hydro generation scheduling, but the precision and efficiency need to further enhance. Hu et al. (2012) combined ant colony algorithm with equal incremental method; however, the operation efficiency drastically slows down with the increase in the number of units and parameter configuration may bring premature convergence trouble [36]. Wu (2015) adopts the measure of discrete particle swarm optimization (DPSO) with relevant improvement; however, global convergence is inefficient which makes it incapable of obtaining convincible solutions [37]. The gravitational search algorithm (GSA) proposed by Shen introduces the chaos theory to enhance local search ability, but the overall efficiency and robustness remain to be strengthened [31, 32]. In summary, the intelligent heuristic stochastic algorithms mentioned present outstanding performance on computational accuracy, stability, and speed compared to that in traditional method. However, few methods may encounter premature convergence with limitations in local and global search strategies, making it difficult to find theoretical optimal solution. Therefore, it is significant and challengeable to develop effective method to realize more efficient and economical hydro generation scheduling.The shuffled frog leaping algorithm (SFLA), a stochastic global optimization method, is inspired by predation behavior of frogs in wetland and first brought forward by Eusuff and Lansey in 2006 [38]. It is found that SFLA can guide the search toward global optimum direction by means of the global information exchange and internal communication mechanism [39]. SFLA, which is easy and convenient to code and has less control parameters, is verified to be compatible with handling comprehensive optimization problems, including the nonlinear and high dimensional discrete systems [40, 41](Li et al. 2014; Cao 2014). Li and Gao both adopt SFLA to solve the problem of economic dispatch and unit commitment and put forth targeted improvements to overcome drawback of SFLA, thus enhancing performance in searching and convergence ability. Nonetheless, the local update formula of SFLA still has limitations, leading the search degree of SFLA not enough to find the optimal solution and easy to be premature convergence, especially when all the frogs are concentrated in a local search field. In fact, the research concerning SFLA is still in the initial stage, in both theoretical and practical application especially the STHGS of hydropower station.The paper focuses on effective improvements and strategies to overcome defects and strengthen performance of SFLA. However, the initial and fundamental SFLA is the real code version with continuous variable value, not appropriate for hydropower systems involving the discrete value (0 and 1) which represents online or offline status of hydropower units, respectively. For that reason, it is essential and necessary to transform real coded SFLA into the binary or discrete version BSFLA. Consequently, the transformational BSFLA is suitable for dealing with the 0/1 programming problem, i.e., the discrete space of unit commitment and scheduling. Meanwhile, the real coded RSFLA is designed for the ELD problem. The improvements for BSFLA contained population initialization operated by renewed logistic chaotic map, renewed frog subpopulation division (leader, follower, and mutation frog), the cloud model for enhancing evolution and exploitation capability of leader and mutation frog subpopulations, and improved search strategies for follower frog subpopulations. Furthermore, the self-adaptive activation mechanism is introduced to the RSFLA. In summary, the improved discrete-real coded shuffled frog leaping algorithm (IBR-SFLA) is developed to solve the STHGS problem. The improved IB-SFLA is proposed as external search to optimize the UC subproblem; simultaneously the improved real coded SFLA (IR-SFLA) is as the internal search to handle ELD subproblem. Finally, the case study on Three Gorges hydropower station is used to test overall performance of IBR-SFLA. The simulation results show that it can figure out reliable high-quality solutions and outperform contrastive approaches and algorithms in precision and stability, while costing shorter time under the same operation condition.


## 2. STHGS Problem Formulation

As mentioned above, the model for STHGS includes two submodules: the UC and ELD. The aim of STGHS is to minimize, to the utmost, water consumption with meeting the power load demand, water balance, spinning reserve constraint, and other system constraints simultaneously in the duration of given operation periods, i.e., an hour for interval in a day.


### 2.1. Economic Load Dispatch (ELD)

As one subproblem of STHGS, the ELD tends to pursue the minimum water consumption of generating power in hydropower station. The objective function can be described as(1)min⁡Q=∑i=1nuiQiHi,PiQ denotes the water consumption during operation periods; ui is the ith hydro unit; Pi&Hi represent load for ith unit to burden and water head for ith unit, respectively; QiHi,Pi is the water consumption by ith hydro unit.Specifically, the water consumption function is set as quadratic function through curve fitting method. The function is expressed as(2)QiHi,Pi=ai+biPi+ciPi2a i, bi, ci are coefficients in the fitting quadratic function and other parameters have the same meaning as the former.


### 2.2. Unit Commitment Problem

Reasonable hydropower units arrangement can bring enormous benefits on power generation efficiency. The basic goal of unit commitment is to effectively schedule the on/off state of hydropower units at the cost of minimum water which is consumed by power generation and switch process of startup and shutdown. The ascertained upstream and downstream water level, load demand, and various constraints are also taken into account [1, 30] (Zheng et al. 2013; Chen et al. 2015). Normally, the objective function can be signified as follows.(3)min⁡W=∑t=1T∑i=1NQitHit,Pit·ΔT·uit+uit·1-uit-1·Qup,i+uit-1·1-uit·Qdn,iW denotes total water consumption of hydropower units composed of power generation and startup and shutdown costs of units; Qup,i, Qdn,i are startup and shutdown water consumption, respectively. ΔT is interval, i.e., one hour; T is the total scheduling periods; Hit indicates water head for ith unit at tth interval; Pit denotes power output for ith unit at current tth interval; QitHt,Pit is the outflow of the ith unit at tth interval corresponding to water head Ht and power output Pit; uit valuing 1 and 0, respectively, indicates startup and shutdown state of ith unit.


### 2.3. Constraints

As is mentioned above, there are a number of various hydraulic and electrical constraints in STHGS problem. It can be summarized as follows:① Power Balance Constraint (4) ∑ i = 1 N u i t · P i t = P D tP D t is the power demand at tth interval of hydropower station; uit and Pit have the same meaning as the former.② Spinning Reserve Constraint (5) ∑ i = 1 N u i t · P i , m a x ≥ P D t + P R tP R t denotes spinning reserve at tth interval.③ Hydropower Unit Output Limits (6) P i , m i n ≤ u i t · P i t ≤ P i , m a xP i , m i n and Pi,max signify minimum and maximum load output of ith unit, respectively.④ Prohibited Operating Region Constraint (7) P i , m i n ≤ P i t ≤ P o z i 1 l P o z i m - 1 u ≤ P i t ≤ P o z i m l P o z i r u ≤ P i t ≤ P i , m a x m = 2,3 … , rr represents total prohibited operation zones, with m to number them. (Pozim)l and (Pozim)u are lower and upper limits corresponding to mth prohibited operation zone for the ith unit.⑤ Minimum Up and Down Time Constraint (8) T i , o n t ≥ T i , u p T i , o f f t ≥ T i , d o w nT i , u p and Ti,down are the minimum online (up time) and offline (down time) limits, ensuring stable and safe operation of hydropower units; Ti,ont and Ti,offt denote accumulated up/down time for ith unit that had been online/offline till tth interval.⑥ The Reservoir Storage and Flow Discharge Limits (9) V t m i n ≤ V t ≤ V t m a x Q t m i n ≤ Q t ≤ Q t m a xV t m i n and Vtmax signify the minimum and maximum storage volume of hydropower station. Correspondingly, Qtmin and Qtmax are lower and upper limits of water discharge, respectively.⑦ The Water Balance Constraint (10) V t = V t - 1 + I t - Q t · Δ tV t is the storage volume of reservoir at tth interval; It, Qt are inflow and outflow (water release); Δt is interval.


### 2.4. Influencing Factors in STHGS

#### 2.4.1. The Stable Operation Region

As is demonstrated in Figure1, the operation area of hydropower unit can be classified into three components: prohibition operation, limited operation, and stable operation regions or zones. Constraint ④ mentioned describes the prohibited operation limits of turbine-unit. Enormous vibration and noise will emerge when hydro unit operates in the prohibited operation zone (POZ) which brings great harm to unit security and stability. Therefore, to ensure long-term security and prolong the service life of units, it is vital to regulate unit operation zone and limit its operation within stable operation zone (SOZ). The dark blue area in Figure 2 shows the unit SOZ of Three Gorges left bank (VGS and ALOSTOM) under different water heads.Figure 1
The operation region of left-bank ALSTOM units in Three Gorges hydropower station.Figure 2
The SOZ of left-bank units in Three Gorges hydropower station.


#### 2.4.2. The Unit Operation Efficiency

The unit operation efficiency (UOE), another important turbine-unit characteristic, has a significant impact on generation and total water consumption. Specifically, the relationship between power generation and UOE can be expressed as the following formula:(11)P=9.81ηQHη denotes unit operation efficiency (%). Q is water discharge and H is effective water head. P represents the power output (kW).Under a certain water head, different unit power output corresponds to various efficiency which is shown in Figure3. The curves in Figure 3, displaying UOE in Xiluodu hydropower station, do not present tendency of monotonic increasing but a sharp reduction after the maximum point [42]. The variation trend of curves is consistent with Figure 1. By analyzing the UOE curves under different operation zones in Figure 1, we can safely conclude that a single unit will have a relative higher efficiency when working under the stable operation zone (SOZ) than that in other zones. In this paper, the UOE and SOZ are fully considered into the ELD subproblem to fully fulfill purpose of minimizing water consumption.Figure 3
The unit operation efficiency (UOE) of right-bank unit in Xiluodu hydropower station.


#### 2.4.3. The Unit Water Consumption Rate

The definition of hydropower unit water consumption rate (UWCR) is the water consumption corresponding to unit power output. Generally, it can be acquired by the following formula:(12)C=QiPi,HiPi=1ηHiwhere QiPi,Hi denotes generation flow. C is water consumption rate of ith unit when power output is Pi and net water head is Hi.From formula (12), we can see the UWCR is concerned with net water head Hi and η (UOE), of which differences are great among different units. The UWCR is introduced as an influencing factor, aiming to preferentially start up unit with lower UWCR and shut down unit with higher UWCR. Thus, the water consumption can be saved and spinning reserve constraint (constraint ②) is satisfied simultaneously.


## 2.1. Economic Load Dispatch (ELD)

As one subproblem of STHGS, the ELD tends to pursue the minimum water consumption of generating power in hydropower station. The objective function can be described as(1)min⁡Q=∑i=1nuiQiHi,PiQ denotes the water consumption during operation periods; ui is the ith hydro unit; Pi&Hi represent load for ith unit to burden and water head for ith unit, respectively; QiHi,Pi is the water consumption by ith hydro unit.Specifically, the water consumption function is set as quadratic function through curve fitting method. The function is expressed as(2)QiHi,Pi=ai+biPi+ciPi2a i, bi, ci are coefficients in the fitting quadratic function and other parameters have the same meaning as the former.


## 2.2. Unit Commitment Problem

Reasonable hydropower units arrangement can bring enormous benefits on power generation efficiency. The basic goal of unit commitment is to effectively schedule the on/off state of hydropower units at the cost of minimum water which is consumed by power generation and switch process of startup and shutdown. The ascertained upstream and downstream water level, load demand, and various constraints are also taken into account [1, 30] (Zheng et al. 2013; Chen et al. 2015). Normally, the objective function can be signified as follows.(3)min⁡W=∑t=1T∑i=1NQitHit,Pit·ΔT·uit+uit·1-uit-1·Qup,i+uit-1·1-uit·Qdn,iW denotes total water consumption of hydropower units composed of power generation and startup and shutdown costs of units; Qup,i, Qdn,i are startup and shutdown water consumption, respectively. ΔT is interval, i.e., one hour; T is the total scheduling periods; Hit indicates water head for ith unit at tth interval; Pit denotes power output for ith unit at current tth interval; QitHt,Pit is the outflow of the ith unit at tth interval corresponding to water head Ht and power output Pit; uit valuing 1 and 0, respectively, indicates startup and shutdown state of ith unit.


## 2.3. Constraints

As is mentioned above, there are a number of various hydraulic and electrical constraints in STHGS problem. It can be summarized as follows:① Power Balance Constraint (4) ∑ i = 1 N u i t · P i t = P D tP D t is the power demand at tth interval of hydropower station; uit and Pit have the same meaning as the former.② Spinning Reserve Constraint (5) ∑ i = 1 N u i t · P i , m a x ≥ P D t + P R tP R t denotes spinning reserve at tth interval.③ Hydropower Unit Output Limits (6) P i , m i n ≤ u i t · P i t ≤ P i , m a xP i , m i n and Pi,max signify minimum and maximum load output of ith unit, respectively.④ Prohibited Operating Region Constraint (7) P i , m i n ≤ P i t ≤ P o z i 1 l P o z i m - 1 u ≤ P i t ≤ P o z i m l P o z i r u ≤ P i t ≤ P i , m a x m = 2,3 … , rr represents total prohibited operation zones, with m to number them. (Pozim)l and (Pozim)u are lower and upper limits corresponding to mth prohibited operation zone for the ith unit.⑤ Minimum Up and Down Time Constraint (8) T i , o n t ≥ T i , u p T i , o f f t ≥ T i , d o w nT i , u p and Ti,down are the minimum online (up time) and offline (down time) limits, ensuring stable and safe operation of hydropower units; Ti,ont and Ti,offt denote accumulated up/down time for ith unit that had been online/offline till tth interval.⑥ The Reservoir Storage and Flow Discharge Limits (9) V t m i n ≤ V t ≤ V t m a x Q t m i n ≤ Q t ≤ Q t m a xV t m i n and Vtmax signify the minimum and maximum storage volume of hydropower station. Correspondingly, Qtmin and Qtmax are lower and upper limits of water discharge, respectively.⑦ The Water Balance Constraint (10) V t = V t - 1 + I t - Q t · Δ tV t is the storage volume of reservoir at tth interval; It, Qt are inflow and outflow (water release); Δt is interval.


## 2.4. Influencing Factors in STHGS

### 2.4.1. The Stable Operation Region

As is demonstrated in Figure1, the operation area of hydropower unit can be classified into three components: prohibition operation, limited operation, and stable operation regions or zones. Constraint ④ mentioned describes the prohibited operation limits of turbine-unit. Enormous vibration and noise will emerge when hydro unit operates in the prohibited operation zone (POZ) which brings great harm to unit security and stability. Therefore, to ensure long-term security and prolong the service life of units, it is vital to regulate unit operation zone and limit its operation within stable operation zone (SOZ). The dark blue area in Figure 2 shows the unit SOZ of Three Gorges left bank (VGS and ALOSTOM) under different water heads.Figure 1
The operation region of left-bank ALSTOM units in Three Gorges hydropower station.Figure 2
The SOZ of left-bank units in Three Gorges hydropower station.


### 2.4.2. The Unit Operation Efficiency

The unit operation efficiency (UOE), another important turbine-unit characteristic, has a significant impact on generation and total water consumption. Specifically, the relationship between power generation and UOE can be expressed as the following formula:(11)P=9.81ηQHη denotes unit operation efficiency (%). Q is water discharge and H is effective water head. P represents the power output (kW).Under a certain water head, different unit power output corresponds to various efficiency which is shown in Figure3. The curves in Figure 3, displaying UOE in Xiluodu hydropower station, do not present tendency of monotonic increasing but a sharp reduction after the maximum point [42]. The variation trend of curves is consistent with Figure 1. By analyzing the UOE curves under different operation zones in Figure 1, we can safely conclude that a single unit will have a relative higher efficiency when working under the stable operation zone (SOZ) than that in other zones. In this paper, the UOE and SOZ are fully considered into the ELD subproblem to fully fulfill purpose of minimizing water consumption.Figure 3
The unit operation efficiency (UOE) of right-bank unit in Xiluodu hydropower station.


### 2.4.3. The Unit Water Consumption Rate

The definition of hydropower unit water consumption rate (UWCR) is the water consumption corresponding to unit power output. Generally, it can be acquired by the following formula:(12)C=QiPi,HiPi=1ηHiwhere QiPi,Hi denotes generation flow. C is water consumption rate of ith unit when power output is Pi and net water head is Hi.From formula (12), we can see the UWCR is concerned with net water head Hi and η (UOE), of which differences are great among different units. The UWCR is introduced as an influencing factor, aiming to preferentially start up unit with lower UWCR and shut down unit with higher UWCR. Thus, the water consumption can be saved and spinning reserve constraint (constraint ②) is satisfied simultaneously.


## 2.4.1. The Stable Operation Region

As is demonstrated in Figure1, the operation area of hydropower unit can be classified into three components: prohibition operation, limited operation, and stable operation regions or zones. Constraint ④ mentioned describes the prohibited operation limits of turbine-unit. Enormous vibration and noise will emerge when hydro unit operates in the prohibited operation zone (POZ) which brings great harm to unit security and stability. Therefore, to ensure long-term security and prolong the service life of units, it is vital to regulate unit operation zone and limit its operation within stable operation zone (SOZ). The dark blue area in Figure 2 shows the unit SOZ of Three Gorges left bank (VGS and ALOSTOM) under different water heads.Figure 1
The operation region of left-bank ALSTOM units in Three Gorges hydropower station.Figure 2
The SOZ of left-bank units in Three Gorges hydropower station.


## 2.4.2. The Unit Operation Efficiency

The unit operation efficiency (UOE), another important turbine-unit characteristic, has a significant impact on generation and total water consumption. Specifically, the relationship between power generation and UOE can be expressed as the following formula:(11)P=9.81ηQHη denotes unit operation efficiency (%). Q is water discharge and H is effective water head. P represents the power output (kW).Under a certain water head, different unit power output corresponds to various efficiency which is shown in Figure3. The curves in Figure 3, displaying UOE in Xiluodu hydropower station, do not present tendency of monotonic increasing but a sharp reduction after the maximum point [42]. The variation trend of curves is consistent with Figure 1. By analyzing the UOE curves under different operation zones in Figure 1, we can safely conclude that a single unit will have a relative higher efficiency when working under the stable operation zone (SOZ) than that in other zones. In this paper, the UOE and SOZ are fully considered into the ELD subproblem to fully fulfill purpose of minimizing water consumption.Figure 3
The unit operation efficiency (UOE) of right-bank unit in Xiluodu hydropower station.


## 2.4.3. The Unit Water Consumption Rate

The definition of hydropower unit water consumption rate (UWCR) is the water consumption corresponding to unit power output. Generally, it can be acquired by the following formula:(12)C=QiPi,HiPi=1ηHiwhere QiPi,Hi denotes generation flow. C is water consumption rate of ith unit when power output is Pi and net water head is Hi.From formula (12), we can see the UWCR is concerned with net water head Hi and η (UOE), of which differences are great among different units. The UWCR is introduced as an influencing factor, aiming to preferentially start up unit with lower UWCR and shut down unit with higher UWCR. Thus, the water consumption can be saved and spinning reserve constraint (constraint ②) is satisfied simultaneously.


## 3. The Introduction of IBR-SFLA and Model Establishment for STHGS Problem

### 3.1. The Overview of Traditional SFLA

Inspired by foraging behavior about frogs in the swamp, the SFLA is a metaheuristic optimization based on swarm intelligence, combining deterministic method with random measure. It incorporates advantage of memetics evolution and particle swarm algorithm which possesses characteristic of less parameters and easiness to comprehension [43] (Zhao and Chen 2016). The frog population is partitioned into a number of parallel subpopulations (communities), of which frogs are arranged to search prey and evolve independently, i.e., memetic evolution. Frogs in each subpopulation can be seen as potential feasible solutions and idea toward a goal and are infected by ideas of other frogs [44] (Zou et al. 2012). Memetic evolution aims to improve quality of individual frog and enhance performance, thus, directing frogs to exploit more excellent ideas and solutions. After a number of memetic evolution steps, the shuffling process, i.e., mingling frogs from different subpopulations, is activated to boost frog quality after exchanging information with each subpopulation so as to pass down the superior ideas [45](Luo et al. 2015).Generally, the frog population division strategy can be described as follows: to begin with, the initial population will be ranked in descending order based on fitness of each frog. Then, the frogs are grouped intom subpopulations which satisfy M=m×h; i.e., each subpopulation contains h frogs. Finally, the first frog will be assessed in first subpopulation and the next one is chosen to the next subpopulation, and so forth, until the mth frog is grouped in the mth subpopulation. The subpopulations are allocated with frogs in the first round, then the (m+1)th frog is divided into the first subpopulation, the (m+2)th frog is divided into the second subpopulation, and so on [39] (Sun et al. 2016). The rest of subpopulations adopt similar grouping measure to make all frogs classified so as to find the optimal and worst frog ub and uw within each subpopulation as well as the current global optimal frogs ug in the whole population.The local search in each subpopulation merely renovates the worst frog individual in frog subpopulations. Usually, the position of frog is updated by the following formula:(13)di=rand∗ub-uw(14)uwnew=uw+didmin≤di≤dmaxr a n d denotes random function generating the number in the interval [0,1]; di is leap step of frog, i =1,2,3... m; dmin and dmax are minimum and maximum frog leap step, respectively; ub and uw denote the best and worst frog positions of each subpopulation, respectively; uwnew is the new updated worst frog position.The position ofuw will be replaced by the new frog position of uwnew if the fitness of renewed frog uwnew is superior to that of previous one uw. Otherwise, new frog position is acquired once through (13), in which ub is replaced by ug. Then, if the process is still incapable of improving performance of worst frog uw, a new frog uwnew is obtained randomly to displace the worst frog position uw. All frogs are mixed together and rearranged in the frog shuffling process when local search is accomplished in all subpopulations. Then, the local evolution and global shuffling will perform repeatedly until the maximum iterations preset or convergence criteria are satisfied [44].


### 3.2. The Improved Binary Coded SFLA (IB-SFLA)

The UC is a discrete subproblem which involves 0-1 state variables. Therefore, the binary coded SFLA with improvements is adopted to find commitment schedule (unit on/off state).


#### 3.2.1. The Grouping Strategy

Usually, the frogs are sorted in a descending order according to fitness in the traditional SFLA. Thereafter, the frogs are divided into subpopulations in order. Consequently, relatively worst frogs are sorted into last subpopulation. The population grouping limitation makes the worst frog in last subpopulation have difficulty in acquiring the excellent information and improving its quality.To avoid this limitation, the frog grouping process is updated. The frog population is divided into three types of memetic subpopulations: leader frog, follower frog, and mutation frog. As the elite individual in whole population, the leader frog subpopulation plays a role in exploring more superior position. Follower frog tends to share information and mutation frog exploits solution scope of unknown area. Frogs in each type of subpopulation experience evolution with different strategies or methods. Therefore, the relative worst frog can better develop new idea with evolution strategy suitable for itself. Meanwhile, the elite information exchange and communication between parallel memetic subpopulations will be more effective. The grouping method is elaborated as follows:Step 1.
Select the optimal frogs in all subpopulations and calculate average fitness of these frogsF(ubavg).Step 2.
Obtain the optimal frogs whose fitness is superior and inferior toF(ubavg) in all subpopulations. Then figure up average fitness of available frogs F(ubbavg) and F(ubwavg), respectively.Step 3.
Count number of subpopulations in which fitness of the optimal frog is superior toF(ubbavg) and inferior to F(ubwavg), respectively. The number is stored in variables mled and mmua; specifically, mled is the number of leader frog subpopulations and mmua denotes number of mutation frog subpopulations.Step 4.
Set up total number of subpopulationsm. The number of leader and mutation frog subpopulations has been acquired in Step 3. Therefore, the number of follower frog subpopulations is m-mled-mmua.


#### 3.2.2. The Modified Search Strategy for Followers Frog Subpopulation

In the improved binary coded SFLA (IB-SFLA), the frog positionu is discretized and coded with values 1 or 0 (on/off), while the frog leap step d need not be discretized. The values of d determine whether u is 1 or 0 in the form of probability. In this paper, the improved sigmoid function is developed to build relationship between frog position u and leap step d. The function details are defined in (15) and graph of function is shown in Figure 4. (15)uw,t+1newi,j,k=1ifdti,j,k≥0andr≤21+e-dti,j,k-1uw,toldi,j,kortherwise0ifdti,j,k≤0andr≤1-21+e-dti,j,kFigure 4
The graph of improved sigmoid function.d t i , j , k is the leap step of kth unit attached to the worst frog in ith subpopulation at jth period; uw,toldi,j,k is the old location of the worst frog in previous tth iteration; uw,t+1newi,j,k is location of the renewed worst frog in ith subpopulation in the t+1 th iteration.Furthermore, in the frog evolution process of traditional SFLA mentioned above, worst frog in subpopulation is updated primarily according to the best frogub in subpopulation. If the new position of the worst frog is still worse than before, frogs are evolved according to the global optimal frog ug in (13). Under this evolution strategy, the information is so scarce for worst frog that it is difficult to develop new idea to the fullest extent. Therefore, the old position updating formula is modified to fully pass down the information and novel idea of ub and ug simultaneously. The new updating strategy is shown in (16) and conducted if the new frog is not to be improved after traditional evolution process. The frog position evolution is still according to (15). (16)dt+1=β1∗ub-uw+β2∗ug-uwd t + 1 is the renewed frog leap step; ub and uw are location of the best and worst frogs, respectively; ug denotes the best location in whole frog population currently. Both β1 and β2 are rand function which creates random number in the interval of [0,1].


#### 3.2.3. The Elite Evolution Strategy for Leader Frog Subpopulation Based on NCM

The cloud theory was put forward by Li et al. in 1995, which possesses the prominent characteristics of randomness and stable tendency, and it has been proved to avoid the local extreme to a certain degree [46]. In recent years, the cloud model has been introduced and combined with a number of algorithms such as GA, PSO, and evolution algorithm [47–49] (Dai et al. 2007; Zhang et al. 2008; Zhang et al. 2012). The introduction of cloud model algorithm can effectively improve overall search algorithm performance and contribute to overcome the defects on easily falling into premature trap to some extent.According to the principle of sociology, the more excellent individual often appears around the current optimal individual in subpopulations. In other words, the better frog is found with great probability in the vicinity of the current best frog. Therefore, the normal cloud model is introduced into SFLA to seek the potential best frog within a feasible zone and reduce possibility of trapping in the local optimum.Normal cloud model (NCM) is brought forth to describe the uncertain conversion relation of qualitative concept or qualitative knowledge with its quantitative expressions. The cloud model is proposed for handling combination of qualitative and quantitative problem. The model reflects the correlation of randomness and fuzziness, establishing the mapping relation with quantitativeness and qualitativeness.U is defined as the domain which is one-dimensional or multidimensional. C is a qualitative concept corresponding to U. For a certain element x∈U, a random number with stable tendency F=Cx is defined as certainty of x to C. The distribution of x in the domain U is called cloud model and each x is one cloud droplet in the domain space. The expectation Ex, entropy En, and hyperentropy He are used for characterizing the digital features of cloud [50] (Ma et al. 2013). The details on Ex, En, and He are as follows:①Ex is central of all droplets and the most representative cloud droplet for qualitative concept. The optimal frog in leader frog subpopulations is selected as the representative cloud droplet.② The entropyEn denotes the uncertainty measurement of qualitative concept. It is not merely reflecting emergence randomness of cloud droplet but also revealing the correlation of fuzziness and randomness. The scope of generating cloud droplet and its randomness highly depends on the value of En; specifically the larger En is, the more obvious the randomness is and the more broader the scope is. En is valued with the standard deviation of the frog fitness in subpopulation during current iteration.③ The hyperentropyHe determined by randomness and fuzziness represents measurement of entropy, i.e., the entropy of entropy. To boost search randomness in initial iteration stage and maintain search stability in the later stage, He is relevant to En with value of En/5.The process of creating cloud droplet according to basic normal cloud generator is detailed as follows [39, 51] (Ding and Wang 2013; Sun et al. 2016).Step 1.
Generate a normal random numberEnn with expectation En and standard deviation He.Step 2.
Thereafter, generate a normal random numberx as a cloud droplet, which is taken Ex, Enn as expectation and standard deviation, respectively.Step 3.
The calculationEnn and x are plugged into formula F=e-(x-Ex)2/2(Enn)2 to figure up certainty pertaining to the qualitative concept C.Step 4.
Repeat the above-mentioned processes until the cloud droplets satisfy the terminal condition.The optimal individual in leader frog subpopulation is regarded as a normal dropletCloud (Ex,En,He). Then, several cloud droplets are created according to this droplet by normal cloud generator. The number of new created cloud droplets is the same with capacity of subpopulation h. The new droplet replaces original optimal frog if fitness is better than before. Otherwise, the original optimal frog is still to be retained [52] (Zhang and Li 2015). The elite evolution strategy aims to evolve the best individual in subpopulation, while the worst frog adopts the traditional updating mechanism to finish evolution.


#### 3.2.4. The Mutation Strategy for Mutation Frog Subpopulation

There are variation phenomena in the biosphere, which has characteristic of randomness and burstiness. However, the biomutation is beneficial to track the potential optimal individual in the unknown solution space and avoid falling into local optimum to some extent.Through analysis of the grouping strategy, it is known that relative worst frogs are sorted into the mutation subpopulations. In comparison with other frog subpopulations, the mutation frog is weaker than others to make contribution to developing novel idea. Therefore, the mutation frogs are designed to find profitable variation and corresponding update strategy for mutation frogs is adaptively changed using chaotic mapping method.The logistic map is a typical discrete-time dynamical systems, i.e., nonlinear equation. The chaotic sequence is produced by logistic map, which is seemingly chaotic but with regularity and ergodic property. In this paper, the modification is developed for conventional logistic map to strengthen population diversity and distribution uniformity. The chaotic sequence is calculated by the following reformative logistic map equation [53] (Zou et al. 2016):(17)xjt+1=1-2xjt2xjt∈-1,1;t=1,2…tmaxj represents decision variable dimension; t is the current iteration; tmax is maximum iterations; variable xjt is defined as the jth chaos variable after tth iteration, valuing in the interval [-1,1].Each dimension of chaotic variablesxj is reflected to feasible solution space of optimization variable, i.e., frog step dij by (18). Then, the conversion relation between mutation frog leap step and position u is established through (15).(18)dij=xjt+12∗dmax--dmin+-dminIf the new positionunew of mutation frog obtains a better performance than the overall optimal frog, the position of current optimal frog is replaced by this position. Otherwise, calculations in (17) and (18) are repeated until the given terminal condition is satisfied. The new individual created by mutation strategy based on chaotic mapping possesses characteristic of randomness, fuzziness, and determinacy. This mutation strategy contributes to effectively exploit space beyond local convergence domain, thus guiding algorithm to locate the more outstanding frog.


### 3.3. The Improved Real Coded SFLA (IR-SFLA)

The ELD subproblem is a continuous, highly nonlinear, and multidimensional optimization problem, aiming at efficiently dispatching power load of hydropower units. An optimal dispatch schedule is beneficial to minimize water consumption. Because the power load is continuous, the real coded mechanism of SFLA is suitable for handling ELD subproblem. However, the evolution strategy in traditional SFLA is liable to slow down convergence speed and not easily escape from local optimum. Thus, several changes for the original real coded version of SFLA (IR-SFLA) are introduced in this paper.


#### 3.3.1. Initial Population Based on Chaos Theory

The initialization has a significant effect on convergence speed and final solution quality. Traditional population initialization is accomplished through the random guided search strategy, making it difficult to locate feasible zone. Chaos theory possesses inherent stochastic characteristic and ergodicity, which is highly sensitive to initial value and liable to escape from local optimum. Therefore, the improved chaos logistic map is applied to frog population initialization to strengthen population diversity and distribution uniformity during the initialization process. It is beneficial to promote overall quality of initial population and locate feasible search zone effectively [53, 54] (Cheng et al. 2008; Zou et al. 2016).Similarly, the chaotic sequencexj creation process is according to (17), and chaotic variables are mapped to feasible solution space of variable yij on the basis of (18). In IR-SFLA, variable yij is real coded, which denotes the power load in ELD subproblem and the frog position updating formula is in line with (14). Then, calculate fitness f(yij) of yij which is concerned with water consumption. If f(yij)k≤f(yij)k+1, i.e., the new position is superior to the old created in the last generation, then f(yij)k=f(yij)k+1 and yijk=yijk+1. The process is repeated in a number of iterations and select the optimal frogs into initial population.


#### 3.3.2. The Improved Search and Position Evolution Strategy

As mentioned in IB-SFLA, the information is too scarce to guide the update of worst frog with using search strategy of traditional SFLA. Similarly, the information of local optimal frogub and overall optimal frog ug is combined to lead the worst frog into novel position. Furthermore, drawing on the PSO, the history frog leap step is taken into consideration to further enrich update information of worst frog. The improved search strategy is shown in the following equation, and the position updating is according to (14). (19)dt+1=θdt+β1∗ub-uw+β2∗ug-uwu b, uw, ug are real coded with continuous variable, i.e., unit power load in ELD. Similarly, dt denotes historical frog leap step; β1 and β2 are evenly distributed random number between 0,1; θ is adjustment coefficient, which aims at controlling leap step and convergence speed in order to achieve balance between local deep-searching and global information shuffling process. In this paper, the improved adjustment coefficient equation is shown in (20)θ=θmax-θminγ∗Lg-LtLg+1-γ∗1-t/Ls1+t/Ls+θminL t is the number of global shuffling iterations; Lg is total global shuffling iterations; t is the number of local iterations in memetic subpopulations; Ls represents total local iterations in subpopulations; θmax, θmin are the maximum and minimum adjustment coefficient; γ is weight coefficient in the interval of 0,1.At the initial evolution stage, the adjustment coefficientθ is assigned with a larger value to boost search capacity in global solution space and a lower value at the later evolution stage. A lower adjustment coefficient θ slows down the evolution speed and promotes depth search in local feasible solution space in the later stage [55].


#### 3.3.3. The Adaptive Frog Activation Mechanism

The local optimal individualub, overall optimal ug, and worst frog uw to be updated are very close with great possibility if frog population search and evolve in a narrow space, i.e., local optimum. Assuming that the difference of (ug - ub) is close to zero, population diversity is reduced and new frog position update for uw will weaken and even stagnate. Therefore, the adaptive frog activation mechanism is introduced into frog update and position evolution process. The activation mechanism replaces inactive frog lack of evolution ability with creating a new individual. Consequently, search ability of uw is reactivated and new position tends to be found. The new frog creation method is expounded as follows.(21)newuw=uw1+θ·β,ifFug-Fub≤εθ=θe+arctan⁡1.561-t∗LtLs∗Lgφ∗θs-θeβ is random number in the range of 0,1; θ denotes adjustment coefficient which uses the arc tangent function to make it changeable dynamically; θs and θe are initial and final value of θ which are evaluated with 0.9 and 0.4, respectively; φ represents controlling factor in the range of [0.4,0.7]; t, Lt, Ls, and Lg are defined in (20). To maintain fast convergence characteristic of IR-SFLA and reduce uncertainty of random number, a relative lower θ is used in the later iteration stage. F(ug)-F(ub)≤ε is decision condition; ε is a positive number defined as similarity threshold; F(ug) and F(ub) are fitness of ug and ub, respectively. Equation (21) is activated if difference of F(ug) and F(ub) satisfies the decision condition in a certain number of iterations.


#### 3.3.4. The Numerical Simulation Experiment

Numerical simulation experiment is designed to demonstrate the superiority of IR-SFLA in solving continuous space optimization problem. The typical Sphere, Rosenbrock, Rastrigin, Griewank, and Schaffer benchmark problems with 30 dimensions are selected as simulation experiment. The IR-SFLA is compared with PPNGA, EACO, SFLA, CGSA, and IBCO, and 20 independent simulations are conducted for all algorithms. Partial results of the 20 simulations are shown in Figure5. The Mean and Standard Deviation (STD) acquired from 20 independent simulations are listed in Table 1, respectively. In addition, the Success Rate and Mean Number of Iteration Generation (MNIG) required to achieve convergence precision preset are introduced to compare convergence speed and reliability of each algorithm, respectively. The results are also listed in Table 1 where the “NA” denotes “Not Applicable” and the convergence is set as 10−8 for all the benchmarks. To further compare IR-SFLA with other rivals, the Friedman and Kruskal-Wallis tests are used to depict the algorithm superiority and listed in Table 2.Table 1
The optimization solutions obtained from IR-SFLA and compared algorithms.
Function PPNGA EACO SFLA CGSA IBCO IR-SFLA Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Sphere 2.82E-71 2.50E-71 2.74E-152 5.42E-152 9.92E-31 1.29E-30 4.90E-51 8.26E-52 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Rosenbrock 1.37E-02 2.12E-03 1.20E+00 1.21E+00 1.00E+00 1.83E+00 4.52E+01 3.78E+01 6.52E+01 8.31E+00 0.00E+00 0.00E+00  Rastrigin 1.09E+00 1.03E+00 3.46E-09 3.09E-09 6.96E-09 1.01E-08 4.01E-07 3.52E-07 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Griewank 0.00E+00 0.00E+00 9.93E-03 1.59E-03 5.07E-03 6.03E-03 2.35E-09 2.84E-09 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Schaffer 1.69E-17 2.63E-17 6.95E-06 1.55E-06 1.72E-13 1.54E-13 4.77E-03 9.50E-03 3.49E-06 5.57E-07 0.00E+00 0.00E+00  Function PPNGA EACO SFLA CGSA IBCO 本文方法 MNIG SR MNIG SR MNIG SR MNIG SR MNIG SR MNIG SR  Sphere 181 100% 313 100% 865 100% 438 100% 82 100% 58 100%  Rosenbrock NA 0% NA 0% NA 0% NA 0% NA 0% 372 100%  Rastrigin NA 0% 1745 95% 2189 90% NA 0% 894 100% 113 100%  Griewank 881 100% NA 0% NA 0% 1794 95% 705 100% 105 100%  Schaffer 482 100% NA 0% 1058 100% NA 0% NA 0% 127 100% Best results are marked in bold.Table 2
The ranks obtained by nonparametric test of Friedman and Kruskal-Wallis for IR-SFLA and compared algorithms.
Nonparametric Test Algorithms PPNGA EACO SFLA CGSA IBCO IR-SFLA Friedman (Rank) 3.42 4.17 3.96 5.02 3.00 1.45  Kruskal-Wallis (Rank) 58.94 73.235 69.865 91.13 50.43 19.4 Best results are marked in bold.Figure 5
The iterative curve of IR-SFLA and compared algorithms for test benchmark functions. Notes: the algorithms convergent to the optimal result “0” after stagnation point.Through inspection of the iteration curves in Figure5 and results of Mean and STD in Table 1, it is clearly revealed that IR-SFLA eventually converges to the evidently beat solution for each benchmark problem in comparison with other algorithms. Furthermore, the performance on convergence speed for IR-SFLA is much better than competitors especially in terms of F2 Rosenbrock, F3 Rastrigin, and F4 Griewank. For F1 Sphere, F3 Rastrigin, and F4 Griewank, the performance for IBCO is outstanding and converges to the best solution; however, the convergence speed for IBCO has no advantage compared with IR-SFLA through the iteration curves in Figure 5. Moreover, the IBCO is not skilled in solving F2 and F5 benchmarks from the aspects of final precision and stability reflected by STD in comparison with IR-SFLA and PPNGA. The method PPNGA presents instable performances which are reflected in relative good performance on F2 and F5, especially the F4 benchmarks and poor performance on F3.From the MNIG and SR in the bottom of Table1, the SR of IR-SFLA are all “100%” which denotes IR-SFLA can obtain the final theoretical optimum of each benchmark in 20 independent simulations which verify the high reliability standard of IR-SFLA. Furthermore, the MNIG of IR-SFLA are the minimum within all the algorithms. However, from the observation of compared algorithms, although the IBCO can get the best results in F1, F3, and F4, its MNIG are much bigger than that in IR-SFLA that indicates the lower convergence speed than the latter. The PPNGA performs well in F4 but is difficult to converge to optimum of other benchmarks. The rest of compared algorithms can not acquire the theoretical optimal solutions of each benchmark but perform better if the precision demand is relatively low. Therefore, the IR-SFLA displays the remarkable advantages on reliability and convergence speed compared to others in terms of F1 and F5 benchmarks.In terms of test ranks from Table2, the less test result is, the better performance algorithm has. It can be seen that IR-SFLA is well positioned among all algorithms which denotes better performance and significant advantages on computational precision and reliability in comparison with other algorithms.From the results summarized above, it can be concluded that the improvements for SFLA present efficient effects on overall search capability and keep a fast convergence speed compared with other rivals. To sum up, the IR-SFLA is more adaptive and effective in solving the optimization problem in continuous space.


### 3.4. The IB-SFLA to Solve UC Subproblem

#### 3.4.1. Structure and Initialization of Frog Individual for UC

The frog individual of UC subproblem represents on or off (1 or 0) state of thekth unit at jth interval. Therefore, the structure of individual for UC can be described as an integer-matrix Uj,k with value 0 or 1 and shown as follows:(22)Uj,k=u1,1u2,1u3,1⋯uk,1u1,2u2,2u3,2⋯uk,2u1,3u2,3u3,3⋯uk,3⋮⋮⋮⋮⋮u1,ju2,ju3,j⋯uk,jwhere k and j are the number of units and operation duration, respectively; the upper limit of j is 24, i.e., a day; uk,j represents the state of kth unit at jth interval.u k , j is initialized with 0 or 1 at each period through following (23).(23)uk,j=1ifrand>0.50otherwisewhere uk,j is the same meaning with that in (22); rand denotes random number between 0 and 1.


#### 3.4.2. Strategies for Handling the Constraints

(1) The Supplement and Repair Strategy for Unit Reserve Constraint. There are two cases during the process of optimal load distribution between each unit. Case 1 is that the total maximum load between all operating units ∑i=1kuiPi,max is less than the sum of requirement Nt,plan and reserve capacity PRt; Case 2 is the total minimum load of all operating units; ∑i=1kuiPi,min is more than load requirement Nt,plan and reserve capacity PRt.In terms of the Cases mentioned above, a dynamic processing method based on priority list of unit startup and shutdown is proposed. The priority list is ranked in ascending order according to the average unit water consumption rate (AUWCR). The AUWCR calculation formula is shown in(24)Favg=∑i=1kQiPi/Piki=1,2,3⋯,kF a v g denotes the average rate of water consumption; Pi is the distributed power load within stable operation region (SOR); Qi(Pi) represents the corresponding water consumption; k is maximum power load within SOR, setting 1MW as an interval.If Case 1 is activated, turn up the current offline units satisfying the minimum down time constraint in turn according to ascending order in the priority list until the reserve capacity constraint is satisfied. Otherwise, shut down the online units satisfying the minimum operation time constraint in turn according to descending order in the priority list until the reserve capacity constraint is satisfied.(2) The Strategy for Repairing Minimum Up and Down Time Constraint. To satisfy the minimum up and down time constraint, a repair strategy is conducted to adjust the units on and off state. The variables Opentimei,j,k and Closetimei,j,k are introduced to record startup and shutdown duration of each unit attached to ith frog at jth period. The record process is described in (25). Meanwhile, the startup and shutdown durations stored in Opentimei,j,k and Closetimei,j,k are verified whether the minimum up and down time constraint is satisfied or not. If not, unit state remains unchanged; otherwise, the frog can realize the state transition in line with (15). (25)Opentimei,j,k=Opentimei,j-1,k+1ifui,j,k=10ifui,j,k=0Closetimei,j,k=Closetimei,j-1,k+1ifui,j,k=00ifui,j,k=1


### 3.5. The IR-SFLA to Solve ELD Subproblem Based on Optimal Economic Operation Table

A wise and feasible unit commitment is acquired with handling the UC subproblem, and proposed IR-SFLA is applied to realize highly efficient load distribution between committed units.


#### 3.5.1. Structure and Initialization of ELD

The structure of ELD is correlated with that in UC, which consists of power output for all committed hydropower units. The specific structure is illustrated as follows:(26)Pj,k=u1,1P1,1u2,1P2,1u3,1P3,1⋯uk,1Pk,1u1,2P1,2u2,2P2,2u3,2P3,2⋯uk,2Pk,2u1,3P1,3u2,3P2,3u3,3P3,3⋯uk,3Pk,3⋮⋮⋮⋮u1,jP1,ju2,jP2,ju3,jP3,j⋯uk,jPk,jwhere Pk,j represents the power output for the kth unit at jth interval; the meaning of uk,j is in line with that in UC structure.Different with UC initialization, the ELD is real-coded continuity optimization problem. Therefore, initialization process of ELD is according to method based on chaotic theory. The logistic mapping equation is in line with (17); however, the reflection equation is demonstrated as follows with corresponding changes according to (18).(27)Pj,k=xjt+12∗Pk,max-Pk,min+Pk,minx j t denotes the chaotic variable generated by improved logistic mapping formula; Pk,min and Pk,max are lower and upper limit for kth unit.


#### 3.5.2. The Optimal Economic Operation Table

The economic operation for large scale hydropower station, especially in terms of Three Gorges hydropower station, is multidimensional, complex nonlinear optimization problem. In order to simplify the solving process and significantly improve the computation efficiency, the optimal economic operation table is established by the proposed IR-SFLA. The table formulation is on premise that all power units are involved in power generation; i.e., all units are startup. In addition, the SOR of units is taken into consideration to keep unit operated in high efficiency and safe region. The established economic operation table is stored in database, thus making it easier to call the table. The solving formula is demonstrated as follows:(28)N1,Q1N1,H=0Ni=Ni+Pi,jQiNi,H=QiNi,H+qi,jFitbestQiNi,H=min⁡QiNi,HN 1 , Q 1 ( N 1 , H ) = 0 are boundary conditions in the initial stage; Ni is accumulative load of units; Qi(Ni,H) denotes water consumption of power generation when total load and water head are Ni and H, respectively; Fitbest(Qi(Ni,H)) is the best frog individual.


#### 3.5.3. The Process of Invoking Optimal Economic Operation Table

As mentioned above, the optimal economic operation table is stored in database so as to be conveniently invoked. The number of startup units is recorded using the variablem when a unit commitment is obtained by the UC submodule. The variable l is defined as a count variable to locate the lth line in optimal economic operation table. The load distribution between units is obtained by locating the specified line in optimal economic operation table according to the load demand Nt,plan corresponding to each interval. The linear interpolation method is adopted to acquire the load distribution if it cannot be located directly in the table. The flowchart of invoking optimal economic operation table is demonstrated in Figure 6 and the detailed procedures are shown as follows.Figure 6
The flowchart of invoking optimal economic operation table.Step 1.
Seek the first line (l=1) of economic operation table and figure up the total load NN1 of m operational units according to (29). Compare NN1 with load demand and judge whether it is submitted to NN1=Nt,plan or not. If not, l=l+1 and turn to Step 2. Otherwise, terminate the search and (30) is adopted to obtain the load distribution of operational units and corresponding water consumption. (29)NN1=∑k=1mNk∗l,k=1,2…m(30)Nk=Nk∗l,k=1,2,3…m
N k ∗ ( l ) denotes load distribution of kth operational unit in lth line of optimal economic operation table; NN1 is total load distribution of all the operational units in lth line. Nk is intermediate variable to store Nk∗(l).Step 2.
CalculateNN1 and NN2 in line with (31). NN1, NN2 are the total load distribution of operational units corresponding to l-1 th and lth lines in optimal economic operation table. Then go to Step 3.(31)NN1=∑k=1mNk∗l-1,k=1,2…mNN2=∑k=1mNk∗l,k=1,2…mStep 3.
Adjustl according to l=l+1 and turn to Step 2 if the criterion NN2< Nt,plan is satisfied. Otherwise, continue to judge whether the criterion NN2= Nt,plan is satisfied; if it is, the power load of each operational unit is obtained by Nk=Nk∗(l), k=1,2,3....m. Otherwise, the linear interpolation of (32) is applied to obtain load distribution of operational units. (32)Nk=Nk∗l-1+Nt,plan-NN1NN2-NN1Nk∗l-Nk∗l-1k=1,2…mStep 4.
Calculate water consumed by power generation and startup and shutdown transition between hydropower units based on NHQ curve.


### 3.6. The Coupled External and Internal Model for STHGS Problem

The STHGS problem consists of the unit commitment (UC) and load distribution subproblems. Therefore, the coupling model consists of the UC (external submodel) and ELD (internal submodel). The UC submodel delivers message to the ELD submodel in order to realize optimal load distribution and compute total water consumption. On the contrary, the information about water consumption due to power generation and unit startup and shutdown pass back to the UC submodel. The specific steps are listed as follows.Step 1.
Initialize the parameters of IBR-SFLA, including number of frog populationsM, subpopulations m, range of frog leap step, inertia weights ωst and ωen, threshold of heuristic frog activation mechanism, iterations in subpopulations, and whole populations Ls and Lg.Step 2.
Generate initial frog population by (23) where uk,j is initialized with 0 (shutdown) or 1 (startup) at each period using the rand function.Step 3.
Adopt the supplement and repair strategy of unit reserve capacity to satisfy the sinning reserve constraints. Record the duration of unit startup and shutdown according to (25). Regulate the unit state at each period with the repair strategy of minimum startup and shutdown time in order to satisfy constraint ⑤.Step 4.
Invoke and seek the optimal economic operation table to obtain load distribution of operational units and total water consumption including electricity generation and startup and shutdown of units. Calculate fitness of frogs which are ranked in descending order.Step 5.
The frog grouping strategy is applied to divide population into leader frogs, followers frogs, and mutation frogs subpopulations.Step 6.
The leader frogs subpopulation adopts the elite evolution strategy to search and update optimal frog in subpopulation while the worst frog renewal in subpopulations is conducted through traditional frog local search strategy; mutation frogs are designed to exploit potential global optimal frog in feasible region on the basis of mutation strategy; the followers frogs update is based on modified search strategy. The process of handling constraints is still according to strategies in Step 3. Then, finish the load distribution through the optimal economic operation table mentioned in Step 4. All subpopulations belonging to each type of frog are updated with corresponding strategy mentioned and renew the local best frogub and global best ug in each iteration.
Judge whether the iteration of local search is up to the number preset or not; if it is, turn to Step 7. Otherwise, go back toStep 6 and continue to search until the local update is finished in all subpopulations.Step 7.
All subpopulations will be mixed to exchange information and idea between different subpopulations. Record the best frog, minimum water consumption, and optimal unit commitment and load distribution in current global shuffling iteration.
Judge whether the global shuffling iteration is achieved or not; if not,turn toStep 5. If it is, the global best frog, unit commitment, and load distribution are found and output the final results.The flowchart of coupling external and internal model for STHGS is demonstrated in Figure7.Figure 7
The flowchart of coupled external and internal model based on IBR-SFLA for solving STHGS.


## 3.1. The Overview of Traditional SFLA

Inspired by foraging behavior about frogs in the swamp, the SFLA is a metaheuristic optimization based on swarm intelligence, combining deterministic method with random measure. It incorporates advantage of memetics evolution and particle swarm algorithm which possesses characteristic of less parameters and easiness to comprehension [43] (Zhao and Chen 2016). The frog population is partitioned into a number of parallel subpopulations (communities), of which frogs are arranged to search prey and evolve independently, i.e., memetic evolution. Frogs in each subpopulation can be seen as potential feasible solutions and idea toward a goal and are infected by ideas of other frogs [44] (Zou et al. 2012). Memetic evolution aims to improve quality of individual frog and enhance performance, thus, directing frogs to exploit more excellent ideas and solutions. After a number of memetic evolution steps, the shuffling process, i.e., mingling frogs from different subpopulations, is activated to boost frog quality after exchanging information with each subpopulation so as to pass down the superior ideas [45](Luo et al. 2015).Generally, the frog population division strategy can be described as follows: to begin with, the initial population will be ranked in descending order based on fitness of each frog. Then, the frogs are grouped intom subpopulations which satisfy M=m×h; i.e., each subpopulation contains h frogs. Finally, the first frog will be assessed in first subpopulation and the next one is chosen to the next subpopulation, and so forth, until the mth frog is grouped in the mth subpopulation. The subpopulations are allocated with frogs in the first round, then the (m+1)th frog is divided into the first subpopulation, the (m+2)th frog is divided into the second subpopulation, and so on [39] (Sun et al. 2016). The rest of subpopulations adopt similar grouping measure to make all frogs classified so as to find the optimal and worst frog ub and uw within each subpopulation as well as the current global optimal frogs ug in the whole population.The local search in each subpopulation merely renovates the worst frog individual in frog subpopulations. Usually, the position of frog is updated by the following formula:(13)di=rand∗ub-uw(14)uwnew=uw+didmin≤di≤dmaxr a n d denotes random function generating the number in the interval [0,1]; di is leap step of frog, i =1,2,3... m; dmin and dmax are minimum and maximum frog leap step, respectively; ub and uw denote the best and worst frog positions of each subpopulation, respectively; uwnew is the new updated worst frog position.The position ofuw will be replaced by the new frog position of uwnew if the fitness of renewed frog uwnew is superior to that of previous one uw. Otherwise, new frog position is acquired once through (13), in which ub is replaced by ug. Then, if the process is still incapable of improving performance of worst frog uw, a new frog uwnew is obtained randomly to displace the worst frog position uw. All frogs are mixed together and rearranged in the frog shuffling process when local search is accomplished in all subpopulations. Then, the local evolution and global shuffling will perform repeatedly until the maximum iterations preset or convergence criteria are satisfied [44].


## 3.2. The Improved Binary Coded SFLA (IB-SFLA)

The UC is a discrete subproblem which involves 0-1 state variables. Therefore, the binary coded SFLA with improvements is adopted to find commitment schedule (unit on/off state).


### 3.2.1. The Grouping Strategy

Usually, the frogs are sorted in a descending order according to fitness in the traditional SFLA. Thereafter, the frogs are divided into subpopulations in order. Consequently, relatively worst frogs are sorted into last subpopulation. The population grouping limitation makes the worst frog in last subpopulation have difficulty in acquiring the excellent information and improving its quality.To avoid this limitation, the frog grouping process is updated. The frog population is divided into three types of memetic subpopulations: leader frog, follower frog, and mutation frog. As the elite individual in whole population, the leader frog subpopulation plays a role in exploring more superior position. Follower frog tends to share information and mutation frog exploits solution scope of unknown area. Frogs in each type of subpopulation experience evolution with different strategies or methods. Therefore, the relative worst frog can better develop new idea with evolution strategy suitable for itself. Meanwhile, the elite information exchange and communication between parallel memetic subpopulations will be more effective. The grouping method is elaborated as follows:Step 1.
Select the optimal frogs in all subpopulations and calculate average fitness of these frogsF(ubavg).Step 2.
Obtain the optimal frogs whose fitness is superior and inferior toF(ubavg) in all subpopulations. Then figure up average fitness of available frogs F(ubbavg) and F(ubwavg), respectively.Step 3.
Count number of subpopulations in which fitness of the optimal frog is superior toF(ubbavg) and inferior to F(ubwavg), respectively. The number is stored in variables mled and mmua; specifically, mled is the number of leader frog subpopulations and mmua denotes number of mutation frog subpopulations.Step 4.
Set up total number of subpopulationsm. The number of leader and mutation frog subpopulations has been acquired in Step 3. Therefore, the number of follower frog subpopulations is m-mled-mmua.


### 3.2.2. The Modified Search Strategy for Followers Frog Subpopulation

In the improved binary coded SFLA (IB-SFLA), the frog positionu is discretized and coded with values 1 or 0 (on/off), while the frog leap step d need not be discretized. The values of d determine whether u is 1 or 0 in the form of probability. In this paper, the improved sigmoid function is developed to build relationship between frog position u and leap step d. The function details are defined in (15) and graph of function is shown in Figure 4. (15)uw,t+1newi,j,k=1ifdti,j,k≥0andr≤21+e-dti,j,k-1uw,toldi,j,kortherwise0ifdti,j,k≤0andr≤1-21+e-dti,j,kFigure 4
The graph of improved sigmoid function.d t i , j , k is the leap step of kth unit attached to the worst frog in ith subpopulation at jth period; uw,toldi,j,k is the old location of the worst frog in previous tth iteration; uw,t+1newi,j,k is location of the renewed worst frog in ith subpopulation in the t+1 th iteration.Furthermore, in the frog evolution process of traditional SFLA mentioned above, worst frog in subpopulation is updated primarily according to the best frogub in subpopulation. If the new position of the worst frog is still worse than before, frogs are evolved according to the global optimal frog ug in (13). Under this evolution strategy, the information is so scarce for worst frog that it is difficult to develop new idea to the fullest extent. Therefore, the old position updating formula is modified to fully pass down the information and novel idea of ub and ug simultaneously. The new updating strategy is shown in (16) and conducted if the new frog is not to be improved after traditional evolution process. The frog position evolution is still according to (15). (16)dt+1=β1∗ub-uw+β2∗ug-uwd t + 1 is the renewed frog leap step; ub and uw are location of the best and worst frogs, respectively; ug denotes the best location in whole frog population currently. Both β1 and β2 are rand function which creates random number in the interval of [0,1].


### 3.2.3. The Elite Evolution Strategy for Leader Frog Subpopulation Based on NCM

The cloud theory was put forward by Li et al. in 1995, which possesses the prominent characteristics of randomness and stable tendency, and it has been proved to avoid the local extreme to a certain degree [46]. In recent years, the cloud model has been introduced and combined with a number of algorithms such as GA, PSO, and evolution algorithm [47–49] (Dai et al. 2007; Zhang et al. 2008; Zhang et al. 2012). The introduction of cloud model algorithm can effectively improve overall search algorithm performance and contribute to overcome the defects on easily falling into premature trap to some extent.According to the principle of sociology, the more excellent individual often appears around the current optimal individual in subpopulations. In other words, the better frog is found with great probability in the vicinity of the current best frog. Therefore, the normal cloud model is introduced into SFLA to seek the potential best frog within a feasible zone and reduce possibility of trapping in the local optimum.Normal cloud model (NCM) is brought forth to describe the uncertain conversion relation of qualitative concept or qualitative knowledge with its quantitative expressions. The cloud model is proposed for handling combination of qualitative and quantitative problem. The model reflects the correlation of randomness and fuzziness, establishing the mapping relation with quantitativeness and qualitativeness.U is defined as the domain which is one-dimensional or multidimensional. C is a qualitative concept corresponding to U. For a certain element x∈U, a random number with stable tendency F=Cx is defined as certainty of x to C. The distribution of x in the domain U is called cloud model and each x is one cloud droplet in the domain space. The expectation Ex, entropy En, and hyperentropy He are used for characterizing the digital features of cloud [50] (Ma et al. 2013). The details on Ex, En, and He are as follows:①Ex is central of all droplets and the most representative cloud droplet for qualitative concept. The optimal frog in leader frog subpopulations is selected as the representative cloud droplet.② The entropyEn denotes the uncertainty measurement of qualitative concept. It is not merely reflecting emergence randomness of cloud droplet but also revealing the correlation of fuzziness and randomness. The scope of generating cloud droplet and its randomness highly depends on the value of En; specifically the larger En is, the more obvious the randomness is and the more broader the scope is. En is valued with the standard deviation of the frog fitness in subpopulation during current iteration.③ The hyperentropyHe determined by randomness and fuzziness represents measurement of entropy, i.e., the entropy of entropy. To boost search randomness in initial iteration stage and maintain search stability in the later stage, He is relevant to En with value of En/5.The process of creating cloud droplet according to basic normal cloud generator is detailed as follows [39, 51] (Ding and Wang 2013; Sun et al. 2016).Step 1.
Generate a normal random numberEnn with expectation En and standard deviation He.Step 2.
Thereafter, generate a normal random numberx as a cloud droplet, which is taken Ex, Enn as expectation and standard deviation, respectively.Step 3.
The calculationEnn and x are plugged into formula F=e-(x-Ex)2/2(Enn)2 to figure up certainty pertaining to the qualitative concept C.Step 4.
Repeat the above-mentioned processes until the cloud droplets satisfy the terminal condition.The optimal individual in leader frog subpopulation is regarded as a normal dropletCloud (Ex,En,He). Then, several cloud droplets are created according to this droplet by normal cloud generator. The number of new created cloud droplets is the same with capacity of subpopulation h. The new droplet replaces original optimal frog if fitness is better than before. Otherwise, the original optimal frog is still to be retained [52] (Zhang and Li 2015). The elite evolution strategy aims to evolve the best individual in subpopulation, while the worst frog adopts the traditional updating mechanism to finish evolution.


### 3.2.4. The Mutation Strategy for Mutation Frog Subpopulation

There are variation phenomena in the biosphere, which has characteristic of randomness and burstiness. However, the biomutation is beneficial to track the potential optimal individual in the unknown solution space and avoid falling into local optimum to some extent.Through analysis of the grouping strategy, it is known that relative worst frogs are sorted into the mutation subpopulations. In comparison with other frog subpopulations, the mutation frog is weaker than others to make contribution to developing novel idea. Therefore, the mutation frogs are designed to find profitable variation and corresponding update strategy for mutation frogs is adaptively changed using chaotic mapping method.The logistic map is a typical discrete-time dynamical systems, i.e., nonlinear equation. The chaotic sequence is produced by logistic map, which is seemingly chaotic but with regularity and ergodic property. In this paper, the modification is developed for conventional logistic map to strengthen population diversity and distribution uniformity. The chaotic sequence is calculated by the following reformative logistic map equation [53] (Zou et al. 2016):(17)xjt+1=1-2xjt2xjt∈-1,1;t=1,2…tmaxj represents decision variable dimension; t is the current iteration; tmax is maximum iterations; variable xjt is defined as the jth chaos variable after tth iteration, valuing in the interval [-1,1].Each dimension of chaotic variablesxj is reflected to feasible solution space of optimization variable, i.e., frog step dij by (18). Then, the conversion relation between mutation frog leap step and position u is established through (15).(18)dij=xjt+12∗dmax--dmin+-dminIf the new positionunew of mutation frog obtains a better performance than the overall optimal frog, the position of current optimal frog is replaced by this position. Otherwise, calculations in (17) and (18) are repeated until the given terminal condition is satisfied. The new individual created by mutation strategy based on chaotic mapping possesses characteristic of randomness, fuzziness, and determinacy. This mutation strategy contributes to effectively exploit space beyond local convergence domain, thus guiding algorithm to locate the more outstanding frog.


## 3.2.1. The Grouping Strategy

Usually, the frogs are sorted in a descending order according to fitness in the traditional SFLA. Thereafter, the frogs are divided into subpopulations in order. Consequently, relatively worst frogs are sorted into last subpopulation. The population grouping limitation makes the worst frog in last subpopulation have difficulty in acquiring the excellent information and improving its quality.To avoid this limitation, the frog grouping process is updated. The frog population is divided into three types of memetic subpopulations: leader frog, follower frog, and mutation frog. As the elite individual in whole population, the leader frog subpopulation plays a role in exploring more superior position. Follower frog tends to share information and mutation frog exploits solution scope of unknown area. Frogs in each type of subpopulation experience evolution with different strategies or methods. Therefore, the relative worst frog can better develop new idea with evolution strategy suitable for itself. Meanwhile, the elite information exchange and communication between parallel memetic subpopulations will be more effective. The grouping method is elaborated as follows:Step 1.
Select the optimal frogs in all subpopulations and calculate average fitness of these frogsF(ubavg).Step 2.
Obtain the optimal frogs whose fitness is superior and inferior toF(ubavg) in all subpopulations. Then figure up average fitness of available frogs F(ubbavg) and F(ubwavg), respectively.Step 3.
Count number of subpopulations in which fitness of the optimal frog is superior toF(ubbavg) and inferior to F(ubwavg), respectively. The number is stored in variables mled and mmua; specifically, mled is the number of leader frog subpopulations and mmua denotes number of mutation frog subpopulations.Step 4.
Set up total number of subpopulationsm. The number of leader and mutation frog subpopulations has been acquired in Step 3. Therefore, the number of follower frog subpopulations is m-mled-mmua.


## 3.2.2. The Modified Search Strategy for Followers Frog Subpopulation

In the improved binary coded SFLA (IB-SFLA), the frog positionu is discretized and coded with values 1 or 0 (on/off), while the frog leap step d need not be discretized. The values of d determine whether u is 1 or 0 in the form of probability. In this paper, the improved sigmoid function is developed to build relationship between frog position u and leap step d. The function details are defined in (15) and graph of function is shown in Figure 4. (15)uw,t+1newi,j,k=1ifdti,j,k≥0andr≤21+e-dti,j,k-1uw,toldi,j,kortherwise0ifdti,j,k≤0andr≤1-21+e-dti,j,kFigure 4
The graph of improved sigmoid function.d t i , j , k is the leap step of kth unit attached to the worst frog in ith subpopulation at jth period; uw,toldi,j,k is the old location of the worst frog in previous tth iteration; uw,t+1newi,j,k is location of the renewed worst frog in ith subpopulation in the t+1 th iteration.Furthermore, in the frog evolution process of traditional SFLA mentioned above, worst frog in subpopulation is updated primarily according to the best frogub in subpopulation. If the new position of the worst frog is still worse than before, frogs are evolved according to the global optimal frog ug in (13). Under this evolution strategy, the information is so scarce for worst frog that it is difficult to develop new idea to the fullest extent. Therefore, the old position updating formula is modified to fully pass down the information and novel idea of ub and ug simultaneously. The new updating strategy is shown in (16) and conducted if the new frog is not to be improved after traditional evolution process. The frog position evolution is still according to (15). (16)dt+1=β1∗ub-uw+β2∗ug-uwd t + 1 is the renewed frog leap step; ub and uw are location of the best and worst frogs, respectively; ug denotes the best location in whole frog population currently. Both β1 and β2 are rand function which creates random number in the interval of [0,1].


## 3.2.3. The Elite Evolution Strategy for Leader Frog Subpopulation Based on NCM

The cloud theory was put forward by Li et al. in 1995, which possesses the prominent characteristics of randomness and stable tendency, and it has been proved to avoid the local extreme to a certain degree [46]. In recent years, the cloud model has been introduced and combined with a number of algorithms such as GA, PSO, and evolution algorithm [47–49] (Dai et al. 2007; Zhang et al. 2008; Zhang et al. 2012). The introduction of cloud model algorithm can effectively improve overall search algorithm performance and contribute to overcome the defects on easily falling into premature trap to some extent.According to the principle of sociology, the more excellent individual often appears around the current optimal individual in subpopulations. In other words, the better frog is found with great probability in the vicinity of the current best frog. Therefore, the normal cloud model is introduced into SFLA to seek the potential best frog within a feasible zone and reduce possibility of trapping in the local optimum.Normal cloud model (NCM) is brought forth to describe the uncertain conversion relation of qualitative concept or qualitative knowledge with its quantitative expressions. The cloud model is proposed for handling combination of qualitative and quantitative problem. The model reflects the correlation of randomness and fuzziness, establishing the mapping relation with quantitativeness and qualitativeness.U is defined as the domain which is one-dimensional or multidimensional. C is a qualitative concept corresponding to U. For a certain element x∈U, a random number with stable tendency F=Cx is defined as certainty of x to C. The distribution of x in the domain U is called cloud model and each x is one cloud droplet in the domain space. The expectation Ex, entropy En, and hyperentropy He are used for characterizing the digital features of cloud [50] (Ma et al. 2013). The details on Ex, En, and He are as follows:①Ex is central of all droplets and the most representative cloud droplet for qualitative concept. The optimal frog in leader frog subpopulations is selected as the representative cloud droplet.② The entropyEn denotes the uncertainty measurement of qualitative concept. It is not merely reflecting emergence randomness of cloud droplet but also revealing the correlation of fuzziness and randomness. The scope of generating cloud droplet and its randomness highly depends on the value of En; specifically the larger En is, the more obvious the randomness is and the more broader the scope is. En is valued with the standard deviation of the frog fitness in subpopulation during current iteration.③ The hyperentropyHe determined by randomness and fuzziness represents measurement of entropy, i.e., the entropy of entropy. To boost search randomness in initial iteration stage and maintain search stability in the later stage, He is relevant to En with value of En/5.The process of creating cloud droplet according to basic normal cloud generator is detailed as follows [39, 51] (Ding and Wang 2013; Sun et al. 2016).Step 1.
Generate a normal random numberEnn with expectation En and standard deviation He.Step 2.
Thereafter, generate a normal random numberx as a cloud droplet, which is taken Ex, Enn as expectation and standard deviation, respectively.Step 3.
The calculationEnn and x are plugged into formula F=e-(x-Ex)2/2(Enn)2 to figure up certainty pertaining to the qualitative concept C.Step 4.
Repeat the above-mentioned processes until the cloud droplets satisfy the terminal condition.The optimal individual in leader frog subpopulation is regarded as a normal dropletCloud (Ex,En,He). Then, several cloud droplets are created according to this droplet by normal cloud generator. The number of new created cloud droplets is the same with capacity of subpopulation h. The new droplet replaces original optimal frog if fitness is better than before. Otherwise, the original optimal frog is still to be retained [52] (Zhang and Li 2015). The elite evolution strategy aims to evolve the best individual in subpopulation, while the worst frog adopts the traditional updating mechanism to finish evolution.


## 3.2.4. The Mutation Strategy for Mutation Frog Subpopulation

There are variation phenomena in the biosphere, which has characteristic of randomness and burstiness. However, the biomutation is beneficial to track the potential optimal individual in the unknown solution space and avoid falling into local optimum to some extent.Through analysis of the grouping strategy, it is known that relative worst frogs are sorted into the mutation subpopulations. In comparison with other frog subpopulations, the mutation frog is weaker than others to make contribution to developing novel idea. Therefore, the mutation frogs are designed to find profitable variation and corresponding update strategy for mutation frogs is adaptively changed using chaotic mapping method.The logistic map is a typical discrete-time dynamical systems, i.e., nonlinear equation. The chaotic sequence is produced by logistic map, which is seemingly chaotic but with regularity and ergodic property. In this paper, the modification is developed for conventional logistic map to strengthen population diversity and distribution uniformity. The chaotic sequence is calculated by the following reformative logistic map equation [53] (Zou et al. 2016):(17)xjt+1=1-2xjt2xjt∈-1,1;t=1,2…tmaxj represents decision variable dimension; t is the current iteration; tmax is maximum iterations; variable xjt is defined as the jth chaos variable after tth iteration, valuing in the interval [-1,1].Each dimension of chaotic variablesxj is reflected to feasible solution space of optimization variable, i.e., frog step dij by (18). Then, the conversion relation between mutation frog leap step and position u is established through (15).(18)dij=xjt+12∗dmax--dmin+-dminIf the new positionunew of mutation frog obtains a better performance than the overall optimal frog, the position of current optimal frog is replaced by this position. Otherwise, calculations in (17) and (18) are repeated until the given terminal condition is satisfied. The new individual created by mutation strategy based on chaotic mapping possesses characteristic of randomness, fuzziness, and determinacy. This mutation strategy contributes to effectively exploit space beyond local convergence domain, thus guiding algorithm to locate the more outstanding frog.


## 3.3. The Improved Real Coded SFLA (IR-SFLA)

The ELD subproblem is a continuous, highly nonlinear, and multidimensional optimization problem, aiming at efficiently dispatching power load of hydropower units. An optimal dispatch schedule is beneficial to minimize water consumption. Because the power load is continuous, the real coded mechanism of SFLA is suitable for handling ELD subproblem. However, the evolution strategy in traditional SFLA is liable to slow down convergence speed and not easily escape from local optimum. Thus, several changes for the original real coded version of SFLA (IR-SFLA) are introduced in this paper.


### 3.3.1. Initial Population Based on Chaos Theory

The initialization has a significant effect on convergence speed and final solution quality. Traditional population initialization is accomplished through the random guided search strategy, making it difficult to locate feasible zone. Chaos theory possesses inherent stochastic characteristic and ergodicity, which is highly sensitive to initial value and liable to escape from local optimum. Therefore, the improved chaos logistic map is applied to frog population initialization to strengthen population diversity and distribution uniformity during the initialization process. It is beneficial to promote overall quality of initial population and locate feasible search zone effectively [53, 54] (Cheng et al. 2008; Zou et al. 2016).Similarly, the chaotic sequencexj creation process is according to (17), and chaotic variables are mapped to feasible solution space of variable yij on the basis of (18). In IR-SFLA, variable yij is real coded, which denotes the power load in ELD subproblem and the frog position updating formula is in line with (14). Then, calculate fitness f(yij) of yij which is concerned with water consumption. If f(yij)k≤f(yij)k+1, i.e., the new position is superior to the old created in the last generation, then f(yij)k=f(yij)k+1 and yijk=yijk+1. The process is repeated in a number of iterations and select the optimal frogs into initial population.


### 3.3.2. The Improved Search and Position Evolution Strategy

As mentioned in IB-SFLA, the information is too scarce to guide the update of worst frog with using search strategy of traditional SFLA. Similarly, the information of local optimal frogub and overall optimal frog ug is combined to lead the worst frog into novel position. Furthermore, drawing on the PSO, the history frog leap step is taken into consideration to further enrich update information of worst frog. The improved search strategy is shown in the following equation, and the position updating is according to (14). (19)dt+1=θdt+β1∗ub-uw+β2∗ug-uwu b, uw, ug are real coded with continuous variable, i.e., unit power load in ELD. Similarly, dt denotes historical frog leap step; β1 and β2 are evenly distributed random number between 0,1; θ is adjustment coefficient, which aims at controlling leap step and convergence speed in order to achieve balance between local deep-searching and global information shuffling process. In this paper, the improved adjustment coefficient equation is shown in (20)θ=θmax-θminγ∗Lg-LtLg+1-γ∗1-t/Ls1+t/Ls+θminL t is the number of global shuffling iterations; Lg is total global shuffling iterations; t is the number of local iterations in memetic subpopulations; Ls represents total local iterations in subpopulations; θmax, θmin are the maximum and minimum adjustment coefficient; γ is weight coefficient in the interval of 0,1.At the initial evolution stage, the adjustment coefficientθ is assigned with a larger value to boost search capacity in global solution space and a lower value at the later evolution stage. A lower adjustment coefficient θ slows down the evolution speed and promotes depth search in local feasible solution space in the later stage [55].


### 3.3.3. The Adaptive Frog Activation Mechanism

The local optimal individualub, overall optimal ug, and worst frog uw to be updated are very close with great possibility if frog population search and evolve in a narrow space, i.e., local optimum. Assuming that the difference of (ug - ub) is close to zero, population diversity is reduced and new frog position update for uw will weaken and even stagnate. Therefore, the adaptive frog activation mechanism is introduced into frog update and position evolution process. The activation mechanism replaces inactive frog lack of evolution ability with creating a new individual. Consequently, search ability of uw is reactivated and new position tends to be found. The new frog creation method is expounded as follows.(21)newuw=uw1+θ·β,ifFug-Fub≤εθ=θe+arctan⁡1.561-t∗LtLs∗Lgφ∗θs-θeβ is random number in the range of 0,1; θ denotes adjustment coefficient which uses the arc tangent function to make it changeable dynamically; θs and θe are initial and final value of θ which are evaluated with 0.9 and 0.4, respectively; φ represents controlling factor in the range of [0.4,0.7]; t, Lt, Ls, and Lg are defined in (20). To maintain fast convergence characteristic of IR-SFLA and reduce uncertainty of random number, a relative lower θ is used in the later iteration stage. F(ug)-F(ub)≤ε is decision condition; ε is a positive number defined as similarity threshold; F(ug) and F(ub) are fitness of ug and ub, respectively. Equation (21) is activated if difference of F(ug) and F(ub) satisfies the decision condition in a certain number of iterations.


### 3.3.4. The Numerical Simulation Experiment

Numerical simulation experiment is designed to demonstrate the superiority of IR-SFLA in solving continuous space optimization problem. The typical Sphere, Rosenbrock, Rastrigin, Griewank, and Schaffer benchmark problems with 30 dimensions are selected as simulation experiment. The IR-SFLA is compared with PPNGA, EACO, SFLA, CGSA, and IBCO, and 20 independent simulations are conducted for all algorithms. Partial results of the 20 simulations are shown in Figure5. The Mean and Standard Deviation (STD) acquired from 20 independent simulations are listed in Table 1, respectively. In addition, the Success Rate and Mean Number of Iteration Generation (MNIG) required to achieve convergence precision preset are introduced to compare convergence speed and reliability of each algorithm, respectively. The results are also listed in Table 1 where the “NA” denotes “Not Applicable” and the convergence is set as 10−8 for all the benchmarks. To further compare IR-SFLA with other rivals, the Friedman and Kruskal-Wallis tests are used to depict the algorithm superiority and listed in Table 2.Table 1
The optimization solutions obtained from IR-SFLA and compared algorithms.
Function PPNGA EACO SFLA CGSA IBCO IR-SFLA Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Sphere 2.82E-71 2.50E-71 2.74E-152 5.42E-152 9.92E-31 1.29E-30 4.90E-51 8.26E-52 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Rosenbrock 1.37E-02 2.12E-03 1.20E+00 1.21E+00 1.00E+00 1.83E+00 4.52E+01 3.78E+01 6.52E+01 8.31E+00 0.00E+00 0.00E+00  Rastrigin 1.09E+00 1.03E+00 3.46E-09 3.09E-09 6.96E-09 1.01E-08 4.01E-07 3.52E-07 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Griewank 0.00E+00 0.00E+00 9.93E-03 1.59E-03 5.07E-03 6.03E-03 2.35E-09 2.84E-09 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Schaffer 1.69E-17 2.63E-17 6.95E-06 1.55E-06 1.72E-13 1.54E-13 4.77E-03 9.50E-03 3.49E-06 5.57E-07 0.00E+00 0.00E+00  Function PPNGA EACO SFLA CGSA IBCO 本文方法 MNIG SR MNIG SR MNIG SR MNIG SR MNIG SR MNIG SR  Sphere 181 100% 313 100% 865 100% 438 100% 82 100% 58 100%  Rosenbrock NA 0% NA 0% NA 0% NA 0% NA 0% 372 100%  Rastrigin NA 0% 1745 95% 2189 90% NA 0% 894 100% 113 100%  Griewank 881 100% NA 0% NA 0% 1794 95% 705 100% 105 100%  Schaffer 482 100% NA 0% 1058 100% NA 0% NA 0% 127 100% Best results are marked in bold.Table 2
The ranks obtained by nonparametric test of Friedman and Kruskal-Wallis for IR-SFLA and compared algorithms.
Nonparametric Test Algorithms PPNGA EACO SFLA CGSA IBCO IR-SFLA Friedman (Rank) 3.42 4.17 3.96 5.02 3.00 1.45  Kruskal-Wallis (Rank) 58.94 73.235 69.865 91.13 50.43 19.4 Best results are marked in bold.Figure 5
The iterative curve of IR-SFLA and compared algorithms for test benchmark functions. Notes: the algorithms convergent to the optimal result “0” after stagnation point.Through inspection of the iteration curves in Figure5 and results of Mean and STD in Table 1, it is clearly revealed that IR-SFLA eventually converges to the evidently beat solution for each benchmark problem in comparison with other algorithms. Furthermore, the performance on convergence speed for IR-SFLA is much better than competitors especially in terms of F2 Rosenbrock, F3 Rastrigin, and F4 Griewank. For F1 Sphere, F3 Rastrigin, and F4 Griewank, the performance for IBCO is outstanding and converges to the best solution; however, the convergence speed for IBCO has no advantage compared with IR-SFLA through the iteration curves in Figure 5. Moreover, the IBCO is not skilled in solving F2 and F5 benchmarks from the aspects of final precision and stability reflected by STD in comparison with IR-SFLA and PPNGA. The method PPNGA presents instable performances which are reflected in relative good performance on F2 and F5, especially the F4 benchmarks and poor performance on F3.From the MNIG and SR in the bottom of Table1, the SR of IR-SFLA are all “100%” which denotes IR-SFLA can obtain the final theoretical optimum of each benchmark in 20 independent simulations which verify the high reliability standard of IR-SFLA. Furthermore, the MNIG of IR-SFLA are the minimum within all the algorithms. However, from the observation of compared algorithms, although the IBCO can get the best results in F1, F3, and F4, its MNIG are much bigger than that in IR-SFLA that indicates the lower convergence speed than the latter. The PPNGA performs well in F4 but is difficult to converge to optimum of other benchmarks. The rest of compared algorithms can not acquire the theoretical optimal solutions of each benchmark but perform better if the precision demand is relatively low. Therefore, the IR-SFLA displays the remarkable advantages on reliability and convergence speed compared to others in terms of F1 and F5 benchmarks.In terms of test ranks from Table2, the less test result is, the better performance algorithm has. It can be seen that IR-SFLA is well positioned among all algorithms which denotes better performance and significant advantages on computational precision and reliability in comparison with other algorithms.From the results summarized above, it can be concluded that the improvements for SFLA present efficient effects on overall search capability and keep a fast convergence speed compared with other rivals. To sum up, the IR-SFLA is more adaptive and effective in solving the optimization problem in continuous space.


## 3.3.1. Initial Population Based on Chaos Theory

The initialization has a significant effect on convergence speed and final solution quality. Traditional population initialization is accomplished through the random guided search strategy, making it difficult to locate feasible zone. Chaos theory possesses inherent stochastic characteristic and ergodicity, which is highly sensitive to initial value and liable to escape from local optimum. Therefore, the improved chaos logistic map is applied to frog population initialization to strengthen population diversity and distribution uniformity during the initialization process. It is beneficial to promote overall quality of initial population and locate feasible search zone effectively [53, 54] (Cheng et al. 2008; Zou et al. 2016).Similarly, the chaotic sequencexj creation process is according to (17), and chaotic variables are mapped to feasible solution space of variable yij on the basis of (18). In IR-SFLA, variable yij is real coded, which denotes the power load in ELD subproblem and the frog position updating formula is in line with (14). Then, calculate fitness f(yij) of yij which is concerned with water consumption. If f(yij)k≤f(yij)k+1, i.e., the new position is superior to the old created in the last generation, then f(yij)k=f(yij)k+1 and yijk=yijk+1. The process is repeated in a number of iterations and select the optimal frogs into initial population.


## 3.3.2. The Improved Search and Position Evolution Strategy

As mentioned in IB-SFLA, the information is too scarce to guide the update of worst frog with using search strategy of traditional SFLA. Similarly, the information of local optimal frogub and overall optimal frog ug is combined to lead the worst frog into novel position. Furthermore, drawing on the PSO, the history frog leap step is taken into consideration to further enrich update information of worst frog. The improved search strategy is shown in the following equation, and the position updating is according to (14). (19)dt+1=θdt+β1∗ub-uw+β2∗ug-uwu b, uw, ug are real coded with continuous variable, i.e., unit power load in ELD. Similarly, dt denotes historical frog leap step; β1 and β2 are evenly distributed random number between 0,1; θ is adjustment coefficient, which aims at controlling leap step and convergence speed in order to achieve balance between local deep-searching and global information shuffling process. In this paper, the improved adjustment coefficient equation is shown in (20)θ=θmax-θminγ∗Lg-LtLg+1-γ∗1-t/Ls1+t/Ls+θminL t is the number of global shuffling iterations; Lg is total global shuffling iterations; t is the number of local iterations in memetic subpopulations; Ls represents total local iterations in subpopulations; θmax, θmin are the maximum and minimum adjustment coefficient; γ is weight coefficient in the interval of 0,1.At the initial evolution stage, the adjustment coefficientθ is assigned with a larger value to boost search capacity in global solution space and a lower value at the later evolution stage. A lower adjustment coefficient θ slows down the evolution speed and promotes depth search in local feasible solution space in the later stage [55].


## 3.3.3. The Adaptive Frog Activation Mechanism

The local optimal individualub, overall optimal ug, and worst frog uw to be updated are very close with great possibility if frog population search and evolve in a narrow space, i.e., local optimum. Assuming that the difference of (ug - ub) is close to zero, population diversity is reduced and new frog position update for uw will weaken and even stagnate. Therefore, the adaptive frog activation mechanism is introduced into frog update and position evolution process. The activation mechanism replaces inactive frog lack of evolution ability with creating a new individual. Consequently, search ability of uw is reactivated and new position tends to be found. The new frog creation method is expounded as follows.(21)newuw=uw1+θ·β,ifFug-Fub≤εθ=θe+arctan⁡1.561-t∗LtLs∗Lgφ∗θs-θeβ is random number in the range of 0,1; θ denotes adjustment coefficient which uses the arc tangent function to make it changeable dynamically; θs and θe are initial and final value of θ which are evaluated with 0.9 and 0.4, respectively; φ represents controlling factor in the range of [0.4,0.7]; t, Lt, Ls, and Lg are defined in (20). To maintain fast convergence characteristic of IR-SFLA and reduce uncertainty of random number, a relative lower θ is used in the later iteration stage. F(ug)-F(ub)≤ε is decision condition; ε is a positive number defined as similarity threshold; F(ug) and F(ub) are fitness of ug and ub, respectively. Equation (21) is activated if difference of F(ug) and F(ub) satisfies the decision condition in a certain number of iterations.


## 3.3.4. The Numerical Simulation Experiment

Numerical simulation experiment is designed to demonstrate the superiority of IR-SFLA in solving continuous space optimization problem. The typical Sphere, Rosenbrock, Rastrigin, Griewank, and Schaffer benchmark problems with 30 dimensions are selected as simulation experiment. The IR-SFLA is compared with PPNGA, EACO, SFLA, CGSA, and IBCO, and 20 independent simulations are conducted for all algorithms. Partial results of the 20 simulations are shown in Figure5. The Mean and Standard Deviation (STD) acquired from 20 independent simulations are listed in Table 1, respectively. In addition, the Success Rate and Mean Number of Iteration Generation (MNIG) required to achieve convergence precision preset are introduced to compare convergence speed and reliability of each algorithm, respectively. The results are also listed in Table 1 where the “NA” denotes “Not Applicable” and the convergence is set as 10−8 for all the benchmarks. To further compare IR-SFLA with other rivals, the Friedman and Kruskal-Wallis tests are used to depict the algorithm superiority and listed in Table 2.Table 1
The optimization solutions obtained from IR-SFLA and compared algorithms.
Function PPNGA EACO SFLA CGSA IBCO IR-SFLA Mean STD Mean STD Mean STD Mean STD Mean STD Mean STD Sphere 2.82E-71 2.50E-71 2.74E-152 5.42E-152 9.92E-31 1.29E-30 4.90E-51 8.26E-52 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Rosenbrock 1.37E-02 2.12E-03 1.20E+00 1.21E+00 1.00E+00 1.83E+00 4.52E+01 3.78E+01 6.52E+01 8.31E+00 0.00E+00 0.00E+00  Rastrigin 1.09E+00 1.03E+00 3.46E-09 3.09E-09 6.96E-09 1.01E-08 4.01E-07 3.52E-07 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Griewank 0.00E+00 0.00E+00 9.93E-03 1.59E-03 5.07E-03 6.03E-03 2.35E-09 2.84E-09 0.00E+00 0.00E+00 0.00E+00 0.00E+00  Schaffer 1.69E-17 2.63E-17 6.95E-06 1.55E-06 1.72E-13 1.54E-13 4.77E-03 9.50E-03 3.49E-06 5.57E-07 0.00E+00 0.00E+00  Function PPNGA EACO SFLA CGSA IBCO 本文方法 MNIG SR MNIG SR MNIG SR MNIG SR MNIG SR MNIG SR  Sphere 181 100% 313 100% 865 100% 438 100% 82 100% 58 100%  Rosenbrock NA 0% NA 0% NA 0% NA 0% NA 0% 372 100%  Rastrigin NA 0% 1745 95% 2189 90% NA 0% 894 100% 113 100%  Griewank 881 100% NA 0% NA 0% 1794 95% 705 100% 105 100%  Schaffer 482 100% NA 0% 1058 100% NA 0% NA 0% 127 100% Best results are marked in bold.Table 2
The ranks obtained by nonparametric test of Friedman and Kruskal-Wallis for IR-SFLA and compared algorithms.
Nonparametric Test Algorithms PPNGA EACO SFLA CGSA IBCO IR-SFLA Friedman (Rank) 3.42 4.17 3.96 5.02 3.00 1.45  Kruskal-Wallis (Rank) 58.94 73.235 69.865 91.13 50.43 19.4 Best results are marked in bold.Figure 5
The iterative curve of IR-SFLA and compared algorithms for test benchmark functions. Notes: the algorithms convergent to the optimal result “0” after stagnation point.Through inspection of the iteration curves in Figure5 and results of Mean and STD in Table 1, it is clearly revealed that IR-SFLA eventually converges to the evidently beat solution for each benchmark problem in comparison with other algorithms. Furthermore, the performance on convergence speed for IR-SFLA is much better than competitors especially in terms of F2 Rosenbrock, F3 Rastrigin, and F4 Griewank. For F1 Sphere, F3 Rastrigin, and F4 Griewank, the performance for IBCO is outstanding and converges to the best solution; however, the convergence speed for IBCO has no advantage compared with IR-SFLA through the iteration curves in Figure 5. Moreover, the IBCO is not skilled in solving F2 and F5 benchmarks from the aspects of final precision and stability reflected by STD in comparison with IR-SFLA and PPNGA. The method PPNGA presents instable performances which are reflected in relative good performance on F2 and F5, especially the F4 benchmarks and poor performance on F3.From the MNIG and SR in the bottom of Table1, the SR of IR-SFLA are all “100%” which denotes IR-SFLA can obtain the final theoretical optimum of each benchmark in 20 independent simulations which verify the high reliability standard of IR-SFLA. Furthermore, the MNIG of IR-SFLA are the minimum within all the algorithms. However, from the observation of compared algorithms, although the IBCO can get the best results in F1, F3, and F4, its MNIG are much bigger than that in IR-SFLA that indicates the lower convergence speed than the latter. The PPNGA performs well in F4 but is difficult to converge to optimum of other benchmarks. The rest of compared algorithms can not acquire the theoretical optimal solutions of each benchmark but perform better if the precision demand is relatively low. Therefore, the IR-SFLA displays the remarkable advantages on reliability and convergence speed compared to others in terms of F1 and F5 benchmarks.In terms of test ranks from Table2, the less test result is, the better performance algorithm has. It can be seen that IR-SFLA is well positioned among all algorithms which denotes better performance and significant advantages on computational precision and reliability in comparison with other algorithms.From the results summarized above, it can be concluded that the improvements for SFLA present efficient effects on overall search capability and keep a fast convergence speed compared with other rivals. To sum up, the IR-SFLA is more adaptive and effective in solving the optimization problem in continuous space.


## 3.4. The IB-SFLA to Solve UC Subproblem

### 3.4.1. Structure and Initialization of Frog Individual for UC

The frog individual of UC subproblem represents on or off (1 or 0) state of thekth unit at jth interval. Therefore, the structure of individual for UC can be described as an integer-matrix Uj,k with value 0 or 1 and shown as follows:(22)Uj,k=u1,1u2,1u3,1⋯uk,1u1,2u2,2u3,2⋯uk,2u1,3u2,3u3,3⋯uk,3⋮⋮⋮⋮⋮u1,ju2,ju3,j⋯uk,jwhere k and j are the number of units and operation duration, respectively; the upper limit of j is 24, i.e., a day; uk,j represents the state of kth unit at jth interval.u k , j is initialized with 0 or 1 at each period through following (23).(23)uk,j=1ifrand>0.50otherwisewhere uk,j is the same meaning with that in (22); rand denotes random number between 0 and 1.


### 3.4.2. Strategies for Handling the Constraints

(1) The Supplement and Repair Strategy for Unit Reserve Constraint. There are two cases during the process of optimal load distribution between each unit. Case 1 is that the total maximum load between all operating units ∑i=1kuiPi,max is less than the sum of requirement Nt,plan and reserve capacity PRt; Case 2 is the total minimum load of all operating units; ∑i=1kuiPi,min is more than load requirement Nt,plan and reserve capacity PRt.In terms of the Cases mentioned above, a dynamic processing method based on priority list of unit startup and shutdown is proposed. The priority list is ranked in ascending order according to the average unit water consumption rate (AUWCR). The AUWCR calculation formula is shown in(24)Favg=∑i=1kQiPi/Piki=1,2,3⋯,kF a v g denotes the average rate of water consumption; Pi is the distributed power load within stable operation region (SOR); Qi(Pi) represents the corresponding water consumption; k is maximum power load within SOR, setting 1MW as an interval.If Case 1 is activated, turn up the current offline units satisfying the minimum down time constraint in turn according to ascending order in the priority list until the reserve capacity constraint is satisfied. Otherwise, shut down the online units satisfying the minimum operation time constraint in turn according to descending order in the priority list until the reserve capacity constraint is satisfied.(2) The Strategy for Repairing Minimum Up and Down Time Constraint. To satisfy the minimum up and down time constraint, a repair strategy is conducted to adjust the units on and off state. The variables Opentimei,j,k and Closetimei,j,k are introduced to record startup and shutdown duration of each unit attached to ith frog at jth period. The record process is described in (25). Meanwhile, the startup and shutdown durations stored in Opentimei,j,k and Closetimei,j,k are verified whether the minimum up and down time constraint is satisfied or not. If not, unit state remains unchanged; otherwise, the frog can realize the state transition in line with (15). (25)Opentimei,j,k=Opentimei,j-1,k+1ifui,j,k=10ifui,j,k=0Closetimei,j,k=Closetimei,j-1,k+1ifui,j,k=00ifui,j,k=1


## 3.4.1. Structure and Initialization of Frog Individual for UC

The frog individual of UC subproblem represents on or off (1 or 0) state of thekth unit at jth interval. Therefore, the structure of individual for UC can be described as an integer-matrix Uj,k with value 0 or 1 and shown as follows:(22)Uj,k=u1,1u2,1u3,1⋯uk,1u1,2u2,2u3,2⋯uk,2u1,3u2,3u3,3⋯uk,3⋮⋮⋮⋮⋮u1,ju2,ju3,j⋯uk,jwhere k and j are the number of units and operation duration, respectively; the upper limit of j is 24, i.e., a day; uk,j represents the state of kth unit at jth interval.u k , j is initialized with 0 or 1 at each period through following (23).(23)uk,j=1ifrand>0.50otherwisewhere uk,j is the same meaning with that in (22); rand denotes random number between 0 and 1.


## 3.4.2. Strategies for Handling the Constraints

(1) The Supplement and Repair Strategy for Unit Reserve Constraint. There are two cases during the process of optimal load distribution between each unit. Case 1 is that the total maximum load between all operating units ∑i=1kuiPi,max is less than the sum of requirement Nt,plan and reserve capacity PRt; Case 2 is the total minimum load of all operating units; ∑i=1kuiPi,min is more than load requirement Nt,plan and reserve capacity PRt.In terms of the Cases mentioned above, a dynamic processing method based on priority list of unit startup and shutdown is proposed. The priority list is ranked in ascending order according to the average unit water consumption rate (AUWCR). The AUWCR calculation formula is shown in(24)Favg=∑i=1kQiPi/Piki=1,2,3⋯,kF a v g denotes the average rate of water consumption; Pi is the distributed power load within stable operation region (SOR); Qi(Pi) represents the corresponding water consumption; k is maximum power load within SOR, setting 1MW as an interval.If Case 1 is activated, turn up the current offline units satisfying the minimum down time constraint in turn according to ascending order in the priority list until the reserve capacity constraint is satisfied. Otherwise, shut down the online units satisfying the minimum operation time constraint in turn according to descending order in the priority list until the reserve capacity constraint is satisfied.(2) The Strategy for Repairing Minimum Up and Down Time Constraint. To satisfy the minimum up and down time constraint, a repair strategy is conducted to adjust the units on and off state. The variables Opentimei,j,k and Closetimei,j,k are introduced to record startup and shutdown duration of each unit attached to ith frog at jth period. The record process is described in (25). Meanwhile, the startup and shutdown durations stored in Opentimei,j,k and Closetimei,j,k are verified whether the minimum up and down time constraint is satisfied or not. If not, unit state remains unchanged; otherwise, the frog can realize the state transition in line with (15). (25)Opentimei,j,k=Opentimei,j-1,k+1ifui,j,k=10ifui,j,k=0Closetimei,j,k=Closetimei,j-1,k+1ifui,j,k=00ifui,j,k=1


## 3.5. The IR-SFLA to Solve ELD Subproblem Based on Optimal Economic Operation Table

A wise and feasible unit commitment is acquired with handling the UC subproblem, and proposed IR-SFLA is applied to realize highly efficient load distribution between committed units.


### 3.5.1. Structure and Initialization of ELD

The structure of ELD is correlated with that in UC, which consists of power output for all committed hydropower units. The specific structure is illustrated as follows:(26)Pj,k=u1,1P1,1u2,1P2,1u3,1P3,1⋯uk,1Pk,1u1,2P1,2u2,2P2,2u3,2P3,2⋯uk,2Pk,2u1,3P1,3u2,3P2,3u3,3P3,3⋯uk,3Pk,3⋮⋮⋮⋮u1,jP1,ju2,jP2,ju3,jP3,j⋯uk,jPk,jwhere Pk,j represents the power output for the kth unit at jth interval; the meaning of uk,j is in line with that in UC structure.Different with UC initialization, the ELD is real-coded continuity optimization problem. Therefore, initialization process of ELD is according to method based on chaotic theory. The logistic mapping equation is in line with (17); however, the reflection equation is demonstrated as follows with corresponding changes according to (18).(27)Pj,k=xjt+12∗Pk,max-Pk,min+Pk,minx j t denotes the chaotic variable generated by improved logistic mapping formula; Pk,min and Pk,max are lower and upper limit for kth unit.


### 3.5.2. The Optimal Economic Operation Table

The economic operation for large scale hydropower station, especially in terms of Three Gorges hydropower station, is multidimensional, complex nonlinear optimization problem. In order to simplify the solving process and significantly improve the computation efficiency, the optimal economic operation table is established by the proposed IR-SFLA. The table formulation is on premise that all power units are involved in power generation; i.e., all units are startup. In addition, the SOR of units is taken into consideration to keep unit operated in high efficiency and safe region. The established economic operation table is stored in database, thus making it easier to call the table. The solving formula is demonstrated as follows:(28)N1,Q1N1,H=0Ni=Ni+Pi,jQiNi,H=QiNi,H+qi,jFitbestQiNi,H=min⁡QiNi,HN 1 , Q 1 ( N 1 , H ) = 0 are boundary conditions in the initial stage; Ni is accumulative load of units; Qi(Ni,H) denotes water consumption of power generation when total load and water head are Ni and H, respectively; Fitbest(Qi(Ni,H)) is the best frog individual.


### 3.5.3. The Process of Invoking Optimal Economic Operation Table

As mentioned above, the optimal economic operation table is stored in database so as to be conveniently invoked. The number of startup units is recorded using the variablem when a unit commitment is obtained by the UC submodule. The variable l is defined as a count variable to locate the lth line in optimal economic operation table. The load distribution between units is obtained by locating the specified line in optimal economic operation table according to the load demand Nt,plan corresponding to each interval. The linear interpolation method is adopted to acquire the load distribution if it cannot be located directly in the table. The flowchart of invoking optimal economic operation table is demonstrated in Figure 6 and the detailed procedures are shown as follows.Figure 6
The flowchart of invoking optimal economic operation table.Step 1.
Seek the first line (l=1) of economic operation table and figure up the total load NN1 of m operational units according to (29). Compare NN1 with load demand and judge whether it is submitted to NN1=Nt,plan or not. If not, l=l+1 and turn to Step 2. Otherwise, terminate the search and (30) is adopted to obtain the load distribution of operational units and corresponding water consumption. (29)NN1=∑k=1mNk∗l,k=1,2…m(30)Nk=Nk∗l,k=1,2,3…m
N k ∗ ( l ) denotes load distribution of kth operational unit in lth line of optimal economic operation table; NN1 is total load distribution of all the operational units in lth line. Nk is intermediate variable to store Nk∗(l).Step 2.
CalculateNN1 and NN2 in line with (31). NN1, NN2 are the total load distribution of operational units corresponding to l-1 th and lth lines in optimal economic operation table. Then go to Step 3.(31)NN1=∑k=1mNk∗l-1,k=1,2…mNN2=∑k=1mNk∗l,k=1,2…mStep 3.
Adjustl according to l=l+1 and turn to Step 2 if the criterion NN2< Nt,plan is satisfied. Otherwise, continue to judge whether the criterion NN2= Nt,plan is satisfied; if it is, the power load of each operational unit is obtained by Nk=Nk∗(l), k=1,2,3....m. Otherwise, the linear interpolation of (32) is applied to obtain load distribution of operational units. (32)Nk=Nk∗l-1+Nt,plan-NN1NN2-NN1Nk∗l-Nk∗l-1k=1,2…mStep 4.
Calculate water consumed by power generation and startup and shutdown transition between hydropower units based on NHQ curve.


## 3.5.1. Structure and Initialization of ELD

The structure of ELD is correlated with that in UC, which consists of power output for all committed hydropower units. The specific structure is illustrated as follows:(26)Pj,k=u1,1P1,1u2,1P2,1u3,1P3,1⋯uk,1Pk,1u1,2P1,2u2,2P2,2u3,2P3,2⋯uk,2Pk,2u1,3P1,3u2,3P2,3u3,3P3,3⋯uk,3Pk,3⋮⋮⋮⋮u1,jP1,ju2,jP2,ju3,jP3,j⋯uk,jPk,jwhere Pk,j represents the power output for the kth unit at jth interval; the meaning of uk,j is in line with that in UC structure.Different with UC initialization, the ELD is real-coded continuity optimization problem. Therefore, initialization process of ELD is according to method based on chaotic theory. The logistic mapping equation is in line with (17); however, the reflection equation is demonstrated as follows with corresponding changes according to (18).(27)Pj,k=xjt+12∗Pk,max-Pk,min+Pk,minx j t denotes the chaotic variable generated by improved logistic mapping formula; Pk,min and Pk,max are lower and upper limit for kth unit.


## 3.5.2. The Optimal Economic Operation Table

The economic operation for large scale hydropower station, especially in terms of Three Gorges hydropower station, is multidimensional, complex nonlinear optimization problem. In order to simplify the solving process and significantly improve the computation efficiency, the optimal economic operation table is established by the proposed IR-SFLA. The table formulation is on premise that all power units are involved in power generation; i.e., all units are startup. In addition, the SOR of units is taken into consideration to keep unit operated in high efficiency and safe region. The established economic operation table is stored in database, thus making it easier to call the table. The solving formula is demonstrated as follows:(28)N1,Q1N1,H=0Ni=Ni+Pi,jQiNi,H=QiNi,H+qi,jFitbestQiNi,H=min⁡QiNi,HN 1 , Q 1 ( N 1 , H ) = 0 are boundary conditions in the initial stage; Ni is accumulative load of units; Qi(Ni,H) denotes water consumption of power generation when total load and water head are Ni and H, respectively; Fitbest(Qi(Ni,H)) is the best frog individual.


## 3.5.3. The Process of Invoking Optimal Economic Operation Table

As mentioned above, the optimal economic operation table is stored in database so as to be conveniently invoked. The number of startup units is recorded using the variablem when a unit commitment is obtained by the UC submodule. The variable l is defined as a count variable to locate the lth line in optimal economic operation table. The load distribution between units is obtained by locating the specified line in optimal economic operation table according to the load demand Nt,plan corresponding to each interval. The linear interpolation method is adopted to acquire the load distribution if it cannot be located directly in the table. The flowchart of invoking optimal economic operation table is demonstrated in Figure 6 and the detailed procedures are shown as follows.Figure 6
The flowchart of invoking optimal economic operation table.Step 1.
Seek the first line (l=1) of economic operation table and figure up the total load NN1 of m operational units according to (29). Compare NN1 with load demand and judge whether it is submitted to NN1=Nt,plan or not. If not, l=l+1 and turn to Step 2. Otherwise, terminate the search and (30) is adopted to obtain the load distribution of operational units and corresponding water consumption. (29)NN1=∑k=1mNk∗l,k=1,2…m(30)Nk=Nk∗l,k=1,2,3…m
N k ∗ ( l ) denotes load distribution of kth operational unit in lth line of optimal economic operation table; NN1 is total load distribution of all the operational units in lth line. Nk is intermediate variable to store Nk∗(l).Step 2.
CalculateNN1 and NN2 in line with (31). NN1, NN2 are the total load distribution of operational units corresponding to l-1 th and lth lines in optimal economic operation table. Then go to Step 3.(31)NN1=∑k=1mNk∗l-1,k=1,2…mNN2=∑k=1mNk∗l,k=1,2…mStep 3.
Adjustl according to l=l+1 and turn to Step 2 if the criterion NN2< Nt,plan is satisfied. Otherwise, continue to judge whether the criterion NN2= Nt,plan is satisfied; if it is, the power load of each operational unit is obtained by Nk=Nk∗(l), k=1,2,3....m. Otherwise, the linear interpolation of (32) is applied to obtain load distribution of operational units. (32)Nk=Nk∗l-1+Nt,plan-NN1NN2-NN1Nk∗l-Nk∗l-1k=1,2…mStep 4.
Calculate water consumed by power generation and startup and shutdown transition between hydropower units based on NHQ curve.


## 3.6. The Coupled External and Internal Model for STHGS Problem

The STHGS problem consists of the unit commitment (UC) and load distribution subproblems. Therefore, the coupling model consists of the UC (external submodel) and ELD (internal submodel). The UC submodel delivers message to the ELD submodel in order to realize optimal load distribution and compute total water consumption. On the contrary, the information about water consumption due to power generation and unit startup and shutdown pass back to the UC submodel. The specific steps are listed as follows.Step 1.
Initialize the parameters of IBR-SFLA, including number of frog populationsM, subpopulations m, range of frog leap step, inertia weights ωst and ωen, threshold of heuristic frog activation mechanism, iterations in subpopulations, and whole populations Ls and Lg.Step 2.
Generate initial frog population by (23) where uk,j is initialized with 0 (shutdown) or 1 (startup) at each period using the rand function.Step 3.
Adopt the supplement and repair strategy of unit reserve capacity to satisfy the sinning reserve constraints. Record the duration of unit startup and shutdown according to (25). Regulate the unit state at each period with the repair strategy of minimum startup and shutdown time in order to satisfy constraint ⑤.Step 4.
Invoke and seek the optimal economic operation table to obtain load distribution of operational units and total water consumption including electricity generation and startup and shutdown of units. Calculate fitness of frogs which are ranked in descending order.Step 5.
The frog grouping strategy is applied to divide population into leader frogs, followers frogs, and mutation frogs subpopulations.Step 6.
The leader frogs subpopulation adopts the elite evolution strategy to search and update optimal frog in subpopulation while the worst frog renewal in subpopulations is conducted through traditional frog local search strategy; mutation frogs are designed to exploit potential global optimal frog in feasible region on the basis of mutation strategy; the followers frogs update is based on modified search strategy. The process of handling constraints is still according to strategies in Step 3. Then, finish the load distribution through the optimal economic operation table mentioned in Step 4. All subpopulations belonging to each type of frog are updated with corresponding strategy mentioned and renew the local best frogub and global best ug in each iteration.
Judge whether the iteration of local search is up to the number preset or not; if it is, turn to Step 7. Otherwise, go back toStep 6 and continue to search until the local update is finished in all subpopulations.Step 7.
All subpopulations will be mixed to exchange information and idea between different subpopulations. Record the best frog, minimum water consumption, and optimal unit commitment and load distribution in current global shuffling iteration.
Judge whether the global shuffling iteration is achieved or not; if not,turn toStep 5. If it is, the global best frog, unit commitment, and load distribution are found and output the final results.The flowchart of coupling external and internal model for STHGS is demonstrated in Figure7.Figure 7
The flowchart of coupled external and internal model based on IBR-SFLA for solving STHGS.


## 4. Case Study

### 4.1. Three Gorges Hydropower Station Description

The Three Gorges hydropower station, the largest hydropower plant in the world, is located on the Yangtze River in China. The hydropower station is initially equipped with 26 mixed-flow hydro units of 700 MW, and 2 power supply hydro units of 50 MW and 6 hydro units with 700MW installed capacity are equipped afterwards. Therefore, the total installed capacity of hydropower station is up to 22,500MW. In this paper, the initial 26 hydro units containing 6 left-bank VGS units, 8 left-bank ALSTOM units, 4 right-bank ORIENT power units, 4 right-bank ALSTOM units, and 4 right-bank HEC units are selected as simulation operation objects.In this study, the 75m, 88m, and 107m water head are selected as three simulation cases and scheduling periods are 24 hours. The stable operation regions (SOZ) of generating units under 75m, 88m, and 107m water head are listed in Table3. Meanwhile, the flow characteristic curves (NHQ curves) of hydro units corresponding to 75m, 88m, and 107m water head are demonstrated in Figure 8. The daily load demands at each interval for hydropower station are drawn in Figure 9.Table 3
Stable operation region of each unit in Three Gorges corresponding to 75m, 88m, and 107m water head (Unit: MW).
Waterhead Left-bankVGS units (1-6#) Left-bankALSTOM units (7-14#) Right-bankORIENT power units (15-18#) Right-bankALSTOM units (19-22#) Right-bankHEC units (23-26#) 75m 410~645 435~640 390~615 430~665 385~575 88m 475~700 525~700 470~700 500~700 465~700 107m 570~700 615~700 585~700 595~700 585~700Figure 8
The NHQ curve of left- and right-bank hydro units corresponding to 75m, 88m, and 107m water head.
(a)
75m (b)
88m (c)
107mFigure 9
The daily load curves at each interval for hydropower station.


### 4.2. Parameter Settings

The parameter settings significantly affect the efficiency of the IB-RSFLA including search ability, stability, and computation speed. In this study, we set a group of parameter combinations and make comparison of simulation results to find the best control parameter combination. The optimal control parameter settings are described as follows:The number of frog populationsM is 80; the number of subpopulations m is 10; the number of frogs in each subpopulation h is 8; the frog leap step is valued in the interval [-4,4]; the value of inertia weights ωst and ωen are, respectively, 0.9 and 0.4; the iterations number of local search in subpopulation Ls is 10 and global search Lg is 50. In addition, the similarity threshold ε used to motivate the heuristic frog activation mechanism is tested to value 10−3.


### 4.3. Results and Analysis

The 75m, 88m, and 107m water heads are selected as the cases and 20 independent simulations are conducted to find the best solution. The unit commitment and economic load dispatch corresponding to the best solution under 75m, 88m, and 107m water head are demonstrated in Table4, respectively. In inspection of economic load dispatch in Table 4, all hydropower units are operated under stable operation region at each interval, and the hydropower unit output limits ③ and prohibited operating region constraint ④ are rigidly satisfied. Furthermore, Table 4 shows the corresponding unit commitment (UC) strictly satisfies minimum up and down time constraint, which illustrate the state of units (startup and shutdown) is reasonably arranged using “strategies for handling the constraints” in Section 3.4.2. The effective UC contributes to avoiding units startup and shutdown frequently and prolonging the service life of hydropower units.Table 4
The best computational solutions of economic load distribution (ELD) and unit commitment (UC) corresponding to 75m, 88m, and 107m water head.
(a)
75m water head
Period(h) Load distribution (MW) Left-bank VGS ALSTOM 1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14#  1 550 550 550 550 550 550 570 570 Down Down Down Down Down Down 2 550 550 550 550 550 550 570 570 Down Down Down Down Down Down 3 560 560 550 550 550 550 570 570 Down Down Down Down Down Down 4 590 590 590 590 590 590 600 600 Down Down Down Down Down Down 5 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 6 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 7 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 8 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 9 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 10 550 550 550 550 550 550 550 550 550 Down Down Down Down Down 11 560 560 560 560 560 560 570 570 570 Down Down Down Down Down 12 550 550 550 550 550 550 570 570 570 Down Down Down Down Down 13 550 550 550 550 550 550 560 550 550 Down Down Down Down Down 14 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 15 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 16 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 17 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 18 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 19 560 560 560 560 560 550 570 570 570 570 Down Down Down Down 20 560 560 560 560 550 550 570 570 Down 570 Down Down Down Down 21 590 590 590 570 570 570 580 580 Down 580 Down Down Down Down 22 590 590 590 570 570 570 580 580 Down 580 Down Down Down Down 23 560 550 550 550 550 550 570 Down Down Down Down Down Down Down 24 560 550 550 550 550 550 570 Down Down Down Down Down Down Down  Right-bank Orient power ALSTOM HEC UC 15# 16# 17# 18# 19# 20# 21# 22# 23# 24# 25# 26#  1 500 Down 500 Down 540 Down Down Down 490 480 480 Down 11111111000000101010001110 2 500 Down 500 Down 540 Down Down Down 490 480 480 Down 11111111000000101010001110 3 520 Down 520 Down 540 Down Down Down 510 510 510 Down 11111111000000101010001110 4 560 Down 540 Down 610 Down Down Down 520 520 520 Down 11111111000000101010001110 5 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111 6 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111 7 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111 8 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111  9 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111 10 490 490 Down 490 510 510 Down Down 470 470 470 470 11111111100000110111001111 11 520 520 Down 520 540 540 Down Down 510 510 510 510 11111111100000110111001111 12 520 500 Down 500 Down 540 Down Down 490 490 490 490 11111111100000110101001111 13 490 490 Down 490 Down 530 Down Down 480 480 480 480 11111111100000110101001111 14 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 15 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 16 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 17 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 18 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 19 520 520 520 520 Down Down Down Down 510 510 510 510 11111111110000111100001111 20 520 520 520 520 Down Down Down Down 510 510 510 510 11111111010000111100001111 21 520 520 520 520 Down 560 Down Down 510 510 510 510 11111111010000111101001111 22 520 520 520 520 Down 560 Down Down 510 510 510 510 11111111010000111101001111 23 520 520 520 Down Down 540 Down Down 510 510 510 490 11111110000000111001001111 24 520 520 520 Down Down 540 Down Down 510 510 510 490 11111110000000111001001111  (b)
88m water head
Period(h) Load distribution (MW) Left-bank VGS ALSTOM 1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14#  1 Down Down Down Down Down Down 640 640 640 635 630 630 630 630 2 Down Down Down Down Down Down 640 640 640 635 630 630 630 630 3 Down Down Down Down Down Down 640 640 640 640 640 640 640 640 4 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 5 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 6 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 7 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 8 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 9 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 10 Down 660 660 Down Down Down 670 670 670 670 670 670 670 670 11 660 660 660 Down Down Down 660 660 660 660 660 660 660 660 12 650 650 650 Down Down Down 660 660 660 660 656 650 650 Down 13 641 640 640 Down Down Down 640 640 640 640 640 640 640 Down 14 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 15 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 16 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 17 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 18 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 19 660 660 660 660 Down Down 660 660 660 660 660 Down 660 Down  20 640 640 640 480 Down Down 639 639 639 639 639 Down 639 Down 21 660 660 660 660 Down Down 670 670 670 670 670 Down 670 Down 22 660 660 660 660 Down Down 670 670 670 670 670 Down 670 Down 23 670 Down Down 670 Down Down 670 670 670 670 670 Down 670 Down 24 670 Down Down 670 Down Down 670 670 670 670 670 Down 670 Down  Right-bank Orient power ALSTOM HEC UC 15# 16# 17# 18# 19# 20# 21# 22# 23# 24# 25# 26#  1 Down Down Down Down 610 610 610 520 Down Down Down Down 00000011111111000011110000 2 Down Down Down Down 610 610 610 520 Down Down Down Down 00000011111111000011110000 3 Down Down Down Down 620 610 610 610 Down Down Down Down 00000011111111000011110000 4 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 5 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 6 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 7 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 8 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 9 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 10 Down Down Down Down 660 660 660 657 Down Down Down Down 01100011111111000011110000 11 Down Down Down Down 634 620 620 620 Down Down Down Down 11100011111111000011110000 12 Down Down Down Down 620 620 620 620 Down Down Down Down 11100011111110000011110000 13 Down Down Down Down 620 620 620 620 Down Down Down Down 11100011111110000011110000 14 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 15 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 16 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 17 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 18 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 19 630 Down Down Down 650 634 620 620 Down Down Down Down 11110011111010100011110000 20 580 Down Down Down 610 610 610 526 Down Down Down Down 11110011111010100011110000 21 630 Down Down Down 660 650 650 650 Down Down Down Down 11110011111010100011110000 22 630 Down Down Down 660 650 650 650 Down Down Down Down 11110011111010100011110000 23 Down Down Down Down 660 660 660 660 Down Down Down Down 10010011111010000011110000 24 Down Down Down Down 660 660 660 660 Down Down Down Down 10010011111010000011110000  (c)
107m water head
Period(h) Load distribution (MW) Left-bank VGS ALSTOM 1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14#  1 590 590 590 590 Down 590 700 Down 700 Down Down 700 Down Down 2 590 590 590 590 Down 590 700 Down 700 Down Down 700 Down Down 3 590 590 590 590 Down 590 700 Down 700 Down Down 700 Down Down 4 700 700 590 590 Down 590 700 Down 700 Down Down 700 Down Down 5 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down 6 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down  7 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down 8 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down 9 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down 10 700 700 590 590 590 590 Down Down Down Down Down 700 Down Down 11 590 590 590 590 590 590 Down Down Down Down Down 700 Down Down 12 590 590 590 590 590 590 Down Down Down Down Down Down Down Down 13 590 590 590 590 590 590 Down Down Down Down Down Down Down Down 14 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 15 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 16 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 17 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 18 590 590 590 590 590 590 Down Down Down Down Down Down Down Down 19 590 590 590 590 590 590 Down Down Down Down Down Down Down Down 20 590 590 572 570 570 570 Down Down Down Down Down Down Down Down 21 700 590 590 590 590 590 Down Down Down Down Down Down Down Down 22 700 590 590 590 590 590 Down Down Down Down Down Down Down Down 23 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 24 700 700 700 590 590 590 Down Down Down Down Down Down Down Down  Right-bank Orient power ALSTOM HEC UC 15# 16# 17# 18# 19# 20# 21# 22# 23# 24# 25# 26#  1 Down Down Down Down Down Down Down Down 605 590 590 590 11110110100100000000001111 2 Down Down Down Down Down Down Down Down 605 590 590 590 11110110100100000000001111 3 Down Down Down Down Down Down Down Down 700 640 590 590 11110110100100000000001111 4 Down Down Down Down Down Down Down Down 700 700 700 630 11110110100100000000001111 5 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 6 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 7 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 8 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 9 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 10 700 700 Down Down Down 700 Down Down 700 700 700 657 11111100000100110001001111 11 700 700 Down 700 Down 700 Down Down 700 700 700 614 11111100000100110101001111 12 700 700 Down 700 Down 700 Down Down 700 700 696 590 11111100000000110101001111 13 700 700 Down 700 Down 700 Down Down 700 661 590 590 11111100000000110101001111 14 700 Down Down 700 Down Down Down Down 700 700 700 630 11111100000000100100001111 15 700 Down 700 Down Down Down Down Down 700 700 700 630 11111100000000101000001111 16 700 Down 700 Down Down Down Down Down 700 700 700 630 11111100000000101000001111 17 700 Down 700 Down Down Down Down Down 700 700 700 630 11111100000000101000001111 18 700 700 700 Down Down Down Down Down 590 590 590 590 11111100000000111000001111 19 700 700 700 700 Down 700 Down Down 700 700 700 614 11111100000000111101001111 20 690 690 690 690 Down 590 Down Down 590 590 590 590 11111100000000111101001111 21 700 700 700 700 Down 700 Down Down 700 700 700 650 11111100000000111101001111 22 700 700 700 700 Down 700 Down Down 700 700 700 650 11111100000000111101001111 23 700 700 Down Down Down Down Down Down 700 700 700 630 11111100000000110000001111 24 700 700 Down Down Down Down Down Down 700 700 700 630 11111100000000110000001111The iterative curves of best and average water consumption at each generation under 75, 88, and 107m water head are shown in Figure10. Furthermore, a bit of “fluctuant points” in the curves denote the possible local optimal point. Before “fluctuant points”, the search by IBR-SFLA is close to standstill state which indicates algorithm trapped in local optimum. However, the search is reactivated and explored further to gradually converge to best solution after “fluctuant points”. The reactivation can be attributed to evolution operator, modified search strategy of followers frog proposed in IBR-SFLA which contribute algorithm to effectively escape from local optimum and motivate overall search capability.Figure 10
The iterative curves of water consumption at each generation corresponding to 75m, 88m, and 107m water head.Table5 gives details of total water consumption at each period corresponding to 75m, 88m, and 107m water head in which the water consumed by power generation and startup and shutdown of units is included.Table 5
The details of water consumption at each interval corresponding to 75m, 88m, and 107m water head.
Time period (h) 75m 88m 107m Water consumption(m3) Water consumption(m3) Water consumption(m3) Power generation Startup & shutdown Total Power generation Startup & shutdown Total Power generation Startup & shutdown Total 1 3.88E+07 2.52E+06 4.13E+07 3.28E+07 2.16E+06 3.50E+07 2.75E+07 2.16E+06 2.96E+07 2 3.88E+07 0.00E+00 3.88E+07 3.28E+07 0.00E+00 3.28E+07 2.75E+07 0.00E+00 2.75E+07 3 3.95E+07 0.00E+00 3.95E+07 3.33E+07 0.00E+00 3.33E+07 2.79E+07 0.00E+00 2.79E+07 4 4.22E+07 0.00E+00 4.22E+07 3.52E+07 0.00E+00 3.52E+07 2.94E+07 0.00E+00 2.94E+07 5 4.17E+07 1.26E+06 4.30E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 7.20E+05 3.01E+07 6 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 7 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 8 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 9 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 10 4.87E+07 5.40E+05 4.92E+07 4.10E+07 3.60E+05 4.14E+07 3.42E+07 3.60E+05 3.45E+07 11 5.09E+07 0.00E+00 5.09E+07 4.29E+07 1.80E+05 4.31E+07 3.58E+07 1.80E+05 3.60E+07 12 4.71E+07 1.80E+05 4.73E+07 3.97E+07 1.80E+05 3.99E+07 3.32E+07 1.80E+05 3.34E+07 13 4.64E+07 0.00E+00 4.64E+07 3.91E+07 0.00E+00 3.91E+07 3.27E+07 0.00E+00 3.27E+07 14 4.17E+07 3.60E+05 4.21E+07 3.52E+07 3.60E+05 3.56E+07 2.93E+07 3.60E+05 2.97E+07 15 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 3.60E+05 2.97E+07 16 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 17 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 18 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.96E+07 1.80E+05 2.98E+07 19 5.09E+07 5.40E+05 5.15E+07 4.29E+07 5.40E+05 4.34E+07 3.58E+07 3.60E+05 3.62E+07 20 4.79E+07 1.80E+05 4.80E+07 4.06E+07 0.00E+00 4.06E+07 3.40E+07 0.00E+00 3.40E+07 21 5.18E+07 1.80E+05 5.20E+07 4.36E+07 0.00E+00 4.36E+07 3.63E+07 0.00E+00 3.63E+07 22 5.18E+07 0.00E+00 5.18E+07 4.36E+07 0.00E+00 4.36E+07 3.63E+07 0.00E+00 3.63E+07 23 4.17E+07 5.40E+05 4.23E+07 3.52E+07 5.40E+05 3.58E+07 2.93E+07 5.40E+05 2.99E+07 24 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 Sum(∑) 1.06E+09 6.30E+06 1.06E+09 8.90E+08 4.32E+06 8.94E+08 7.43E+08 5.40E+06 7.48E+08To verify whether the power balance constraint ① is satisfied, the comparison between daily load demand and real power output is presented in Figure11 which reflects that real power output satisfies the load demand at each interval.Figure 11
The daily load demand and real power output curve of hydropower station in the Three Gorges.In order to profoundly demonstrate the superiority of IBR-SFLA in solving STHGS problem, a comparison experiment between IBR-SFLA and limited adaptive genetic algorithm (LAGA), extended ant colony optimization (EACO), chaotic gravitational search algorithm (CGSA), firefly algorithm (FA), and improved bee colony optimization (IBCO) is carried out under the same independent simulation times, iteration number, and population scale. The final optimal simulation results obtained by IBR-SFLA and other compared algorithms are listed in Figures12 and 13. Figure 12 presents iterative process of total water consumption for each algorithm. From Figure 12, it can be observed the LAGA is less effective than other algorithms. The uncertainty of control parameters in LAGA dramatically affects the quality of solutions while the method IBR-SFLA proposed has relative less parameters we need to control. Results obtained by IBR-SFLA are obviously more superior than others and water consumption descends drastically at the initial intervals and converges to stable optimal level at the later phase. The IBR-SFLA outperforms the other algorithms through inspection of the convergence precision, which indicates the efficiency and adaption of IBR-SFLA in dealing with STHGS problem.Figure 12
The iterative process of water consumption obtained by IBR-SFLA and compared algorithms corresponding to 75m, 88m, and 107m water head.Figure 13
The box plots of water consumption in 20 simulations obtained by IBR-SFLA and compared algorithms.More details will be revealed if the results obtained by each algorithm in 20 independent simulations are expressed with form of box plot (Figure13). The box plots present distribution of the simulation results from which IBR-SFLA exhibits significantly lower and narrower box plot with short whiskers than that in other rivals under 75m and 88m water heads. It can be stated that solutions obtained by IBR-SFLA in 20 simulations are higher quality and closer to the optimum than compared algorithms. In terms of 107 water heads, the boxes obtained using IBR-SFLA are lower than others but are not well at the coverage area of box plot. However, the whisker of box plot for IBR-SFLA is short which indicates that IBR-SFLA outperforms other algorithms in convergence precision and reliability. To sum up, the higher quality results obtained by IBR-SFLA in 20 simulations demonstrate more effective and stable performance in comparison with other methods.The results analysis simulated by each algorithm is listed in Table6, in which the water consumption, standard deviation, and mean simulation time are shown simultaneously. The advantages of the IBR-SFLA compared with rivals are presented in Tables 6 and 7 and Figure 14. In terms of convergence precision, it is clear that IBR-SFLA can seek the less water consumption through more reasonable unit commitment and load dispatch. The results comparison demonstrates that the improved strategies introduced to IBR-SFLA clearly improve the search capability and overall efficiency to converge to better quality solution.Table 6
The water consumption and average computational time of each algorithm corresponding to 75, 88, and 107m water head.
Waterhead Algorithm Water consumption Standard deviation(STD) Simulation time (s) (108m3) Mean Optimal 75m LAGA 1.074E+01 1.073E+01 8.90E-03 224 EACO 1.068E+01 1.067E+01 8.58E-03 105 CGSA 1.067E+01 1.065E+01 1.20E-02 93 FA 1.066E+01 1.065E+01 9.50E-03 76 IBCO 1.066E+01 1.064E+01 8.13E-03 82 IBR-SFLA 1.063E+01 1.062E+01 4.46E-03 64 88m LAGA 9.071E+00 9.057E+00 5.60E-03 218 EACO 9.006E+00 8.994E+00 8.68E-03 100 CGSA 9.000E+00 8.984E+00 9.14E-03 92 FA 8.984E+00 8.971E+00 6.93E-03 75 IBCO 8.992E+00 8.978E+00 8.08E-03 79 IBR-SFLA 8.950E+00 8.944E+00 3.49E-03 63 107m LAGA 7.567E+00 7.552E+00 1.42E-02 220 EACO 7.549E+00 7.533E+00 9.42E-03 102 CGSA 7.543E+00 7.530E+00 8.62E-03 95 FA 7.528E+00 7.510E+00 9.55E-03 78 IBCO 7.537E+00 7.523E+00 8.39E-03 84 IBR-SFLA 7.492E+00 7.482E+00 6.04E-03 66Table 7
The advantage of IBR-SFLA compared between four algorithms corresponding to water head (75, 88, 107m).
Water head Algorithm Algorithm compared Water consumption Simulation time Standard deviation Reduction(108m3) Reduction (%) (STD) Mean Optimum Mean Optimum Reduction(s) Reduction rate Reduction Reduction rate 75m IBR-SFLA LAGA 1.14E-01 1.06E-01 1.06% 0.99% 160 71.43% 4.44E-03 49.93% EACO 5.32E-02 4.74E-02 0.50% 0.44% 41 39.05% 4.12E-03 48.04% CGSA 4.32E-02 3.19E-02 0.40% 0.30% 29 31.18% 7.50E-03 62.72% FA 3.68E-02 2.81E-02 0.35% 0.26% 12 15.79% 5.04E-03 53.07% IBCO 2.99E-02 2.29E-02 0.28% 0.21% 18 21.95% 3.68E-03 45.22% 88m IBR-SFLA LAGA 1.22E-01 1.13E-01 1.34% 1.24% 155 71.10% 2.10E-03 37.58% EACO 5.61E-02 5.01E-02 0.62% 0.56% 37 37.00% 5.19E-03 59.77% CGSA 5.00E-02 3.96E-02 0.56% 0.44% 29 31.52% 5.64E-03 61.77% FA 3.42E-02 2.72E-02 0.38% 0.30% 12 16.00% 3.44E-03 49.61% IBCO 4.24E-02 3.44E-02 0.47% 0.38% 16 20.25% 4.58E-03 56.76% 107m IBR-SFLA LAGA 7.52E-02 6.97E-02 0.99% 0.92% 154 70.00% 8.13E-03 57.37% EACO 5.68E-02 5.05E-02 0.75% 0.67% 36 35.29% 3.37E-03 35.83% CGSA 5.11E-02 4.78E-02 0.68% 0.64% 29 30.53% 2.58E-03 29.91% FA 3.62E-02 2.82E-02 0.48% 0.38% 12 15.38% 3.51E-03 36.76% IBCO 4.44E-02 4.14E-02 0.59% 0.55% 18 21.43% 2.34E-03 27.95%Figure 14
The contrasting result between four algorithms corresponding to three types of water head (75, 88, and 107m).
(a)
75m water head (b)
88m water head (c)
107m water headMoreover, the simulation time cost by IBR-SFLA is observably less than other compared methods that reflect higher efficiency in dealing with STHGS for large scale hydropower station. The relative higher search speed has close relationship with optimizing mechanism of IBR-SFLA. Only the worst frog in frog subpopulations is updated during each iteration; therefore, only one calculation is needed to finish the iteration in the best case. Even if in the worst case, three calculations are required during each iteration.In terms of algorithm stability, it can be concluded that IBR-SFLA has significant advantage over standard deviation (STD) compared with other algorithms. The smaller STD manifests that computational stability of IBR-SFLA is obviously improved and reflected by narrower box and shorter whiskers in Figure13. Consequently, the proposed IBR-SFLA is more likely to converge to final optimal solution in each simulation than other rivals.In summary, IBR-SFLA shows higher solution quality, convergence speed, and reliability and stability in solving STHGS problem with complex hydraulic constraints. Therefore, the proposed IBR-SFLA is verified as a feasible and effective method to save water resources and further enhance waterpower utilization efficiency.


## 4.1. Three Gorges Hydropower Station Description

The Three Gorges hydropower station, the largest hydropower plant in the world, is located on the Yangtze River in China. The hydropower station is initially equipped with 26 mixed-flow hydro units of 700 MW, and 2 power supply hydro units of 50 MW and 6 hydro units with 700MW installed capacity are equipped afterwards. Therefore, the total installed capacity of hydropower station is up to 22,500MW. In this paper, the initial 26 hydro units containing 6 left-bank VGS units, 8 left-bank ALSTOM units, 4 right-bank ORIENT power units, 4 right-bank ALSTOM units, and 4 right-bank HEC units are selected as simulation operation objects.In this study, the 75m, 88m, and 107m water head are selected as three simulation cases and scheduling periods are 24 hours. The stable operation regions (SOZ) of generating units under 75m, 88m, and 107m water head are listed in Table3. Meanwhile, the flow characteristic curves (NHQ curves) of hydro units corresponding to 75m, 88m, and 107m water head are demonstrated in Figure 8. The daily load demands at each interval for hydropower station are drawn in Figure 9.Table 3
Stable operation region of each unit in Three Gorges corresponding to 75m, 88m, and 107m water head (Unit: MW).
Waterhead Left-bankVGS units (1-6#) Left-bankALSTOM units (7-14#) Right-bankORIENT power units (15-18#) Right-bankALSTOM units (19-22#) Right-bankHEC units (23-26#) 75m 410~645 435~640 390~615 430~665 385~575 88m 475~700 525~700 470~700 500~700 465~700 107m 570~700 615~700 585~700 595~700 585~700Figure 8
The NHQ curve of left- and right-bank hydro units corresponding to 75m, 88m, and 107m water head.
(a)
75m (b)
88m (c)
107mFigure 9
The daily load curves at each interval for hydropower station.


## 4.2. Parameter Settings

The parameter settings significantly affect the efficiency of the IB-RSFLA including search ability, stability, and computation speed. In this study, we set a group of parameter combinations and make comparison of simulation results to find the best control parameter combination. The optimal control parameter settings are described as follows:The number of frog populationsM is 80; the number of subpopulations m is 10; the number of frogs in each subpopulation h is 8; the frog leap step is valued in the interval [-4,4]; the value of inertia weights ωst and ωen are, respectively, 0.9 and 0.4; the iterations number of local search in subpopulation Ls is 10 and global search Lg is 50. In addition, the similarity threshold ε used to motivate the heuristic frog activation mechanism is tested to value 10−3.


## 4.3. Results and Analysis

The 75m, 88m, and 107m water heads are selected as the cases and 20 independent simulations are conducted to find the best solution. The unit commitment and economic load dispatch corresponding to the best solution under 75m, 88m, and 107m water head are demonstrated in Table4, respectively. In inspection of economic load dispatch in Table 4, all hydropower units are operated under stable operation region at each interval, and the hydropower unit output limits ③ and prohibited operating region constraint ④ are rigidly satisfied. Furthermore, Table 4 shows the corresponding unit commitment (UC) strictly satisfies minimum up and down time constraint, which illustrate the state of units (startup and shutdown) is reasonably arranged using “strategies for handling the constraints” in Section 3.4.2. The effective UC contributes to avoiding units startup and shutdown frequently and prolonging the service life of hydropower units.Table 4
The best computational solutions of economic load distribution (ELD) and unit commitment (UC) corresponding to 75m, 88m, and 107m water head.
(a)
75m water head
Period(h) Load distribution (MW) Left-bank VGS ALSTOM 1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14#  1 550 550 550 550 550 550 570 570 Down Down Down Down Down Down 2 550 550 550 550 550 550 570 570 Down Down Down Down Down Down 3 560 560 550 550 550 550 570 570 Down Down Down Down Down Down 4 590 590 590 590 590 590 600 600 Down Down Down Down Down Down 5 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 6 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 7 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 8 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 9 560 550 550 550 550 550 Down Down 570 Down Down Down Down Down 10 550 550 550 550 550 550 550 550 550 Down Down Down Down Down 11 560 560 560 560 560 560 570 570 570 Down Down Down Down Down 12 550 550 550 550 550 550 570 570 570 Down Down Down Down Down 13 550 550 550 550 550 550 560 550 550 Down Down Down Down Down 14 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 15 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 16 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 17 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 18 550 550 550 550 550 550 570 Down 570 Down Down Down Down Down 19 560 560 560 560 560 550 570 570 570 570 Down Down Down Down 20 560 560 560 560 550 550 570 570 Down 570 Down Down Down Down 21 590 590 590 570 570 570 580 580 Down 580 Down Down Down Down 22 590 590 590 570 570 570 580 580 Down 580 Down Down Down Down 23 560 550 550 550 550 550 570 Down Down Down Down Down Down Down 24 560 550 550 550 550 550 570 Down Down Down Down Down Down Down  Right-bank Orient power ALSTOM HEC UC 15# 16# 17# 18# 19# 20# 21# 22# 23# 24# 25# 26#  1 500 Down 500 Down 540 Down Down Down 490 480 480 Down 11111111000000101010001110 2 500 Down 500 Down 540 Down Down Down 490 480 480 Down 11111111000000101010001110 3 520 Down 520 Down 540 Down Down Down 510 510 510 Down 11111111000000101010001110 4 560 Down 540 Down 610 Down Down Down 520 520 520 Down 11111111000000101010001110 5 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111 6 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111 7 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111 8 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111  9 520 520 Down 520 540 Down Down Down 510 510 510 490 11111100100000110110001111 10 490 490 Down 490 510 510 Down Down 470 470 470 470 11111111100000110111001111 11 520 520 Down 520 540 540 Down Down 510 510 510 510 11111111100000110111001111 12 520 500 Down 500 Down 540 Down Down 490 490 490 490 11111111100000110101001111 13 490 490 Down 490 Down 530 Down Down 480 480 480 480 11111111100000110101001111 14 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 15 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 16 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 17 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 18 520 520 Down 520 Down Down Down Down 510 510 490 490 11111110100000110100001111 19 520 520 520 520 Down Down Down Down 510 510 510 510 11111111110000111100001111 20 520 520 520 520 Down Down Down Down 510 510 510 510 11111111010000111100001111 21 520 520 520 520 Down 560 Down Down 510 510 510 510 11111111010000111101001111 22 520 520 520 520 Down 560 Down Down 510 510 510 510 11111111010000111101001111 23 520 520 520 Down Down 540 Down Down 510 510 510 490 11111110000000111001001111 24 520 520 520 Down Down 540 Down Down 510 510 510 490 11111110000000111001001111  (b)
88m water head
Period(h) Load distribution (MW) Left-bank VGS ALSTOM 1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14#  1 Down Down Down Down Down Down 640 640 640 635 630 630 630 630 2 Down Down Down Down Down Down 640 640 640 635 630 630 630 630 3 Down Down Down Down Down Down 640 640 640 640 640 640 640 640 4 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 5 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 6 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 7 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 8 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 9 Down Down Down Down Down Down 670 670 670 670 670 670 670 670 10 Down 660 660 Down Down Down 670 670 670 670 670 670 670 670 11 660 660 660 Down Down Down 660 660 660 660 660 660 660 660 12 650 650 650 Down Down Down 660 660 660 660 656 650 650 Down 13 641 640 640 Down Down Down 640 640 640 640 640 640 640 Down 14 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 15 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 16 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 17 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 18 670 670 Down Down Down Down 670 670 670 670 670 Down 670 Down 19 660 660 660 660 Down Down 660 660 660 660 660 Down 660 Down  20 640 640 640 480 Down Down 639 639 639 639 639 Down 639 Down 21 660 660 660 660 Down Down 670 670 670 670 670 Down 670 Down 22 660 660 660 660 Down Down 670 670 670 670 670 Down 670 Down 23 670 Down Down 670 Down Down 670 670 670 670 670 Down 670 Down 24 670 Down Down 670 Down Down 670 670 670 670 670 Down 670 Down  Right-bank Orient power ALSTOM HEC UC 15# 16# 17# 18# 19# 20# 21# 22# 23# 24# 25# 26#  1 Down Down Down Down 610 610 610 520 Down Down Down Down 00000011111111000011110000 2 Down Down Down Down 610 610 610 520 Down Down Down Down 00000011111111000011110000 3 Down Down Down Down 620 610 610 610 Down Down Down Down 00000011111111000011110000 4 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 5 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 6 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 7 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 8 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 9 Down Down Down Down 660 660 660 660 Down Down Down Down 00000011111111000011110000 10 Down Down Down Down 660 660 660 657 Down Down Down Down 01100011111111000011110000 11 Down Down Down Down 634 620 620 620 Down Down Down Down 11100011111111000011110000 12 Down Down Down Down 620 620 620 620 Down Down Down Down 11100011111110000011110000 13 Down Down Down Down 620 620 620 620 Down Down Down Down 11100011111110000011110000 14 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 15 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 16 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 17 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 18 Down Down Down Down 660 660 660 660 Down Down Down Down 11000011111010000011110000 19 630 Down Down Down 650 634 620 620 Down Down Down Down 11110011111010100011110000 20 580 Down Down Down 610 610 610 526 Down Down Down Down 11110011111010100011110000 21 630 Down Down Down 660 650 650 650 Down Down Down Down 11110011111010100011110000 22 630 Down Down Down 660 650 650 650 Down Down Down Down 11110011111010100011110000 23 Down Down Down Down 660 660 660 660 Down Down Down Down 10010011111010000011110000 24 Down Down Down Down 660 660 660 660 Down Down Down Down 10010011111010000011110000  (c)
107m water head
Period(h) Load distribution (MW) Left-bank VGS ALSTOM 1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14#  1 590 590 590 590 Down 590 700 Down 700 Down Down 700 Down Down 2 590 590 590 590 Down 590 700 Down 700 Down Down 700 Down Down 3 590 590 590 590 Down 590 700 Down 700 Down Down 700 Down Down 4 700 700 590 590 Down 590 700 Down 700 Down Down 700 Down Down 5 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down 6 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down  7 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down 8 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down 9 700 700 700 590 590 590 Down Down Down Down Down 700 Down Down 10 700 700 590 590 590 590 Down Down Down Down Down 700 Down Down 11 590 590 590 590 590 590 Down Down Down Down Down 700 Down Down 12 590 590 590 590 590 590 Down Down Down Down Down Down Down Down 13 590 590 590 590 590 590 Down Down Down Down Down Down Down Down 14 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 15 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 16 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 17 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 18 590 590 590 590 590 590 Down Down Down Down Down Down Down Down 19 590 590 590 590 590 590 Down Down Down Down Down Down Down Down 20 590 590 572 570 570 570 Down Down Down Down Down Down Down Down 21 700 590 590 590 590 590 Down Down Down Down Down Down Down Down 22 700 590 590 590 590 590 Down Down Down Down Down Down Down Down 23 700 700 700 590 590 590 Down Down Down Down Down Down Down Down 24 700 700 700 590 590 590 Down Down Down Down Down Down Down Down  Right-bank Orient power ALSTOM HEC UC 15# 16# 17# 18# 19# 20# 21# 22# 23# 24# 25# 26#  1 Down Down Down Down Down Down Down Down 605 590 590 590 11110110100100000000001111 2 Down Down Down Down Down Down Down Down 605 590 590 590 11110110100100000000001111 3 Down Down Down Down Down Down Down Down 700 640 590 590 11110110100100000000001111 4 Down Down Down Down Down Down Down Down 700 700 700 630 11110110100100000000001111 5 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 6 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 7 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 8 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 9 Down Down Down Down Down 700 Down Down 700 700 700 630 11111100000100000001001111 10 700 700 Down Down Down 700 Down Down 700 700 700 657 11111100000100110001001111 11 700 700 Down 700 Down 700 Down Down 700 700 700 614 11111100000100110101001111 12 700 700 Down 700 Down 700 Down Down 700 700 696 590 11111100000000110101001111 13 700 700 Down 700 Down 700 Down Down 700 661 590 590 11111100000000110101001111 14 700 Down Down 700 Down Down Down Down 700 700 700 630 11111100000000100100001111 15 700 Down 700 Down Down Down Down Down 700 700 700 630 11111100000000101000001111 16 700 Down 700 Down Down Down Down Down 700 700 700 630 11111100000000101000001111 17 700 Down 700 Down Down Down Down Down 700 700 700 630 11111100000000101000001111 18 700 700 700 Down Down Down Down Down 590 590 590 590 11111100000000111000001111 19 700 700 700 700 Down 700 Down Down 700 700 700 614 11111100000000111101001111 20 690 690 690 690 Down 590 Down Down 590 590 590 590 11111100000000111101001111 21 700 700 700 700 Down 700 Down Down 700 700 700 650 11111100000000111101001111 22 700 700 700 700 Down 700 Down Down 700 700 700 650 11111100000000111101001111 23 700 700 Down Down Down Down Down Down 700 700 700 630 11111100000000110000001111 24 700 700 Down Down Down Down Down Down 700 700 700 630 11111100000000110000001111The iterative curves of best and average water consumption at each generation under 75, 88, and 107m water head are shown in Figure10. Furthermore, a bit of “fluctuant points” in the curves denote the possible local optimal point. Before “fluctuant points”, the search by IBR-SFLA is close to standstill state which indicates algorithm trapped in local optimum. However, the search is reactivated and explored further to gradually converge to best solution after “fluctuant points”. The reactivation can be attributed to evolution operator, modified search strategy of followers frog proposed in IBR-SFLA which contribute algorithm to effectively escape from local optimum and motivate overall search capability.Figure 10
The iterative curves of water consumption at each generation corresponding to 75m, 88m, and 107m water head.Table5 gives details of total water consumption at each period corresponding to 75m, 88m, and 107m water head in which the water consumed by power generation and startup and shutdown of units is included.Table 5
The details of water consumption at each interval corresponding to 75m, 88m, and 107m water head.
Time period (h) 75m 88m 107m Water consumption(m3) Water consumption(m3) Water consumption(m3) Power generation Startup & shutdown Total Power generation Startup & shutdown Total Power generation Startup & shutdown Total 1 3.88E+07 2.52E+06 4.13E+07 3.28E+07 2.16E+06 3.50E+07 2.75E+07 2.16E+06 2.96E+07 2 3.88E+07 0.00E+00 3.88E+07 3.28E+07 0.00E+00 3.28E+07 2.75E+07 0.00E+00 2.75E+07 3 3.95E+07 0.00E+00 3.95E+07 3.33E+07 0.00E+00 3.33E+07 2.79E+07 0.00E+00 2.79E+07 4 4.22E+07 0.00E+00 4.22E+07 3.52E+07 0.00E+00 3.52E+07 2.94E+07 0.00E+00 2.94E+07 5 4.17E+07 1.26E+06 4.30E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 7.20E+05 3.01E+07 6 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 7 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 8 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 9 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 10 4.87E+07 5.40E+05 4.92E+07 4.10E+07 3.60E+05 4.14E+07 3.42E+07 3.60E+05 3.45E+07 11 5.09E+07 0.00E+00 5.09E+07 4.29E+07 1.80E+05 4.31E+07 3.58E+07 1.80E+05 3.60E+07 12 4.71E+07 1.80E+05 4.73E+07 3.97E+07 1.80E+05 3.99E+07 3.32E+07 1.80E+05 3.34E+07 13 4.64E+07 0.00E+00 4.64E+07 3.91E+07 0.00E+00 3.91E+07 3.27E+07 0.00E+00 3.27E+07 14 4.17E+07 3.60E+05 4.21E+07 3.52E+07 3.60E+05 3.56E+07 2.93E+07 3.60E+05 2.97E+07 15 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 3.60E+05 2.97E+07 16 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 17 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 18 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.96E+07 1.80E+05 2.98E+07 19 5.09E+07 5.40E+05 5.15E+07 4.29E+07 5.40E+05 4.34E+07 3.58E+07 3.60E+05 3.62E+07 20 4.79E+07 1.80E+05 4.80E+07 4.06E+07 0.00E+00 4.06E+07 3.40E+07 0.00E+00 3.40E+07 21 5.18E+07 1.80E+05 5.20E+07 4.36E+07 0.00E+00 4.36E+07 3.63E+07 0.00E+00 3.63E+07 22 5.18E+07 0.00E+00 5.18E+07 4.36E+07 0.00E+00 4.36E+07 3.63E+07 0.00E+00 3.63E+07 23 4.17E+07 5.40E+05 4.23E+07 3.52E+07 5.40E+05 3.58E+07 2.93E+07 5.40E+05 2.99E+07 24 4.17E+07 0.00E+00 4.17E+07 3.52E+07 0.00E+00 3.52E+07 2.93E+07 0.00E+00 2.93E+07 Sum(∑) 1.06E+09 6.30E+06 1.06E+09 8.90E+08 4.32E+06 8.94E+08 7.43E+08 5.40E+06 7.48E+08To verify whether the power balance constraint ① is satisfied, the comparison between daily load demand and real power output is presented in Figure11 which reflects that real power output satisfies the load demand at each interval.Figure 11
The daily load demand and real power output curve of hydropower station in the Three Gorges.In order to profoundly demonstrate the superiority of IBR-SFLA in solving STHGS problem, a comparison experiment between IBR-SFLA and limited adaptive genetic algorithm (LAGA), extended ant colony optimization (EACO), chaotic gravitational search algorithm (CGSA), firefly algorithm (FA), and improved bee colony optimization (IBCO) is carried out under the same independent simulation times, iteration number, and population scale. The final optimal simulation results obtained by IBR-SFLA and other compared algorithms are listed in Figures12 and 13. Figure 12 presents iterative process of total water consumption for each algorithm. From Figure 12, it can be observed the LAGA is less effective than other algorithms. The uncertainty of control parameters in LAGA dramatically affects the quality of solutions while the method IBR-SFLA proposed has relative less parameters we need to control. Results obtained by IBR-SFLA are obviously more superior than others and water consumption descends drastically at the initial intervals and converges to stable optimal level at the later phase. The IBR-SFLA outperforms the other algorithms through inspection of the convergence precision, which indicates the efficiency and adaption of IBR-SFLA in dealing with STHGS problem.Figure 12
The iterative process of water consumption obtained by IBR-SFLA and compared algorithms corresponding to 75m, 88m, and 107m water head.Figure 13
The box plots of water consumption in 20 simulations obtained by IBR-SFLA and compared algorithms.More details will be revealed if the results obtained by each algorithm in 20 independent simulations are expressed with form of box plot (Figure13). The box plots present distribution of the simulation results from which IBR-SFLA exhibits significantly lower and narrower box plot with short whiskers than that in other rivals under 75m and 88m water heads. It can be stated that solutions obtained by IBR-SFLA in 20 simulations are higher quality and closer to the optimum than compared algorithms. In terms of 107 water heads, the boxes obtained using IBR-SFLA are lower than others but are not well at the coverage area of box plot. However, the whisker of box plot for IBR-SFLA is short which indicates that IBR-SFLA outperforms other algorithms in convergence precision and reliability. To sum up, the higher quality results obtained by IBR-SFLA in 20 simulations demonstrate more effective and stable performance in comparison with other methods.The results analysis simulated by each algorithm is listed in Table6, in which the water consumption, standard deviation, and mean simulation time are shown simultaneously. The advantages of the IBR-SFLA compared with rivals are presented in Tables 6 and 7 and Figure 14. In terms of convergence precision, it is clear that IBR-SFLA can seek the less water consumption through more reasonable unit commitment and load dispatch. The results comparison demonstrates that the improved strategies introduced to IBR-SFLA clearly improve the search capability and overall efficiency to converge to better quality solution.Table 6
The water consumption and average computational time of each algorithm corresponding to 75, 88, and 107m water head.
Waterhead Algorithm Water consumption Standard deviation(STD) Simulation time (s) (108m3) Mean Optimal 75m LAGA 1.074E+01 1.073E+01 8.90E-03 224 EACO 1.068E+01 1.067E+01 8.58E-03 105 CGSA 1.067E+01 1.065E+01 1.20E-02 93 FA 1.066E+01 1.065E+01 9.50E-03 76 IBCO 1.066E+01 1.064E+01 8.13E-03 82 IBR-SFLA 1.063E+01 1.062E+01 4.46E-03 64 88m LAGA 9.071E+00 9.057E+00 5.60E-03 218 EACO 9.006E+00 8.994E+00 8.68E-03 100 CGSA 9.000E+00 8.984E+00 9.14E-03 92 FA 8.984E+00 8.971E+00 6.93E-03 75 IBCO 8.992E+00 8.978E+00 8.08E-03 79 IBR-SFLA 8.950E+00 8.944E+00 3.49E-03 63 107m LAGA 7.567E+00 7.552E+00 1.42E-02 220 EACO 7.549E+00 7.533E+00 9.42E-03 102 CGSA 7.543E+00 7.530E+00 8.62E-03 95 FA 7.528E+00 7.510E+00 9.55E-03 78 IBCO 7.537E+00 7.523E+00 8.39E-03 84 IBR-SFLA 7.492E+00 7.482E+00 6.04E-03 66Table 7
The advantage of IBR-SFLA compared between four algorithms corresponding to water head (75, 88, 107m).
Water head Algorithm Algorithm compared Water consumption Simulation time Standard deviation Reduction(108m3) Reduction (%) (STD) Mean Optimum Mean Optimum Reduction(s) Reduction rate Reduction Reduction rate 75m IBR-SFLA LAGA 1.14E-01 1.06E-01 1.06% 0.99% 160 71.43% 4.44E-03 49.93% EACO 5.32E-02 4.74E-02 0.50% 0.44% 41 39.05% 4.12E-03 48.04% CGSA 4.32E-02 3.19E-02 0.40% 0.30% 29 31.18% 7.50E-03 62.72% FA 3.68E-02 2.81E-02 0.35% 0.26% 12 15.79% 5.04E-03 53.07% IBCO 2.99E-02 2.29E-02 0.28% 0.21% 18 21.95% 3.68E-03 45.22% 88m IBR-SFLA LAGA 1.22E-01 1.13E-01 1.34% 1.24% 155 71.10% 2.10E-03 37.58% EACO 5.61E-02 5.01E-02 0.62% 0.56% 37 37.00% 5.19E-03 59.77% CGSA 5.00E-02 3.96E-02 0.56% 0.44% 29 31.52% 5.64E-03 61.77% FA 3.42E-02 2.72E-02 0.38% 0.30% 12 16.00% 3.44E-03 49.61% IBCO 4.24E-02 3.44E-02 0.47% 0.38% 16 20.25% 4.58E-03 56.76% 107m IBR-SFLA LAGA 7.52E-02 6.97E-02 0.99% 0.92% 154 70.00% 8.13E-03 57.37% EACO 5.68E-02 5.05E-02 0.75% 0.67% 36 35.29% 3.37E-03 35.83% CGSA 5.11E-02 4.78E-02 0.68% 0.64% 29 30.53% 2.58E-03 29.91% FA 3.62E-02 2.82E-02 0.48% 0.38% 12 15.38% 3.51E-03 36.76% IBCO 4.44E-02 4.14E-02 0.59% 0.55% 18 21.43% 2.34E-03 27.95%Figure 14
The contrasting result between four algorithms corresponding to three types of water head (75, 88, and 107m).
(a)
75m water head (b)
88m water head (c)
107m water headMoreover, the simulation time cost by IBR-SFLA is observably less than other compared methods that reflect higher efficiency in dealing with STHGS for large scale hydropower station. The relative higher search speed has close relationship with optimizing mechanism of IBR-SFLA. Only the worst frog in frog subpopulations is updated during each iteration; therefore, only one calculation is needed to finish the iteration in the best case. Even if in the worst case, three calculations are required during each iteration.In terms of algorithm stability, it can be concluded that IBR-SFLA has significant advantage over standard deviation (STD) compared with other algorithms. The smaller STD manifests that computational stability of IBR-SFLA is obviously improved and reflected by narrower box and shorter whiskers in Figure13. Consequently, the proposed IBR-SFLA is more likely to converge to final optimal solution in each simulation than other rivals.In summary, IBR-SFLA shows higher solution quality, convergence speed, and reliability and stability in solving STHGS problem with complex hydraulic constraints. Therefore, the proposed IBR-SFLA is verified as a feasible and effective method to save water resources and further enhance waterpower utilization efficiency.


## 5. Conclusions

In this paper, an improved binary-real coded SFLA (IBR-SFLA) is proposed for solving the STHGS problem for large scale hydropower station. To demonstrate the superiority of IBR-SFLA, numerical simulation experiment is conducted and results show IR-SFLA has exceptional performance on search capability and convergence speed. Then, IBR-SFLA is applied to the STHGS problem in Three Gorges hydropower station and heuristic repairing strategies are introduced to handle complex constraints such as minimum startup and shutdown, spinning reserve, and water balance. Simulation results indicate IBR-SFLA can get better water consumption solution with costing less computation time in comparison with compared methods. By further analysis, distribution of solutions obtained by IBR-SFLA is concentrated with less scatter, which verifies that stability and reliability of IBR-SFLA hold superiority compared to other methods. In conclusion, the proposed IBR-SFLA enhances overall performance of traditional SFLA with more reasonable grouping strategy, effective search strategy based on cloud model, and chaotic theory. It is demonstrated that IBR-SFLA provides more effective and feasible way to solve STHGS problem with less water consumption and higher utilization rate of hydropower resources. In the future, related studies are focused on further improving convergence speed and enhancing comprehensive profitability for hydropower.



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*Source: 3726274-2018-08-13.xml*