Optimization of hydraulic parameters for pipeline system of hydropower station with super long headrace tunnel based on mayfly algorithm considering operational scenarios

This paper studies the optimization of hydraulic parameters for pipeline system of hydropower station with super long headrace tunnel (HSSLHT) based on mayfly algorithm considering operational scenarios. Firstly, the state equation of HSSLHT under load disturbance is derived. The optimization design of hydraulic parameters for pipeline system based on mayfly algorithm is proposed. Then, the optimization of hydraulic parameters is conducted and analyzed. Finally, the effects of pipeline diameters and transfer coefficients of turbines on the optimization of hydraulic parameters for pipeline systems are revealed. The results show that the optimization of hydraulic parameters for pipeline system is a multiobjective problem, and the several objective functions exhibit significant conflicts. Compared to the firefly algorithm and genetic algorithm, the objective function under mayfly algorithm is improved by 14.7% and 5.1%, respectively. The mayfly algorithm can make the hydraulic parameters for the pipeline system reach the Pareto optimal solution under both load decrease condition and load increase condition. The diameter of penstock has an obvious influence on the robustness of dynamic performance of HSSLHT. When the diameter of penstock increases by 30%, the robustness of HSSLHT becomes worse and the robustness index deteriorates by 57%. The reason is that the flow inertia of penstock becomes smaller with the increase of diameter, and the flow inertia of penstock is favorable for resisting disturbance of HSSLHT. The coefficient of throttled orifice head loss and the lengths and head losses of headrace tunnel and penstock are the key hydraulic parameters for matching the operation of HSSLHT.

The pipeline system is an important component of hydropower stations.The optimization of hydraulic parameters for pipeline system is the key aspect of the design of hydropower stations. 1,2The hydraulic parameters for pipeline systems directly influence the safety, stability, and economy of the steady processes and transient processes.On the one hand, the pipeline system includes several subcomponents, and each subcomponent has several hydraulic parameters. 3,4All the subcomponents and hydraulic parameters need to be optimized.On the other hand, the optimization of hydraulic parameters for pipeline system relates to several control requirements, which include the water hammer pressure of pipeline system and stability of turbine unit.The different hydraulic parameters have different relational relationships with different control requirements. 5,6The optimization of hydraulic parameters for pipeline system is a coordinated control problem with multivariable and multiobjective.
Among the different types of hydropower stations, the hydropower station with super long headrace tunnel (HSSLHT) obtains sufficient head by using a super long headrace tunnel (SLHT). 7,8Because of the good adaptability to the complex topographic conditions, the HSSLHT has received more and more attention.For the HSSLHT, the flow inertia of SLHT is great, and the nonlinearity of head loss of SLHT is strong.0][11][12] The great flow inertia of SLHT and huge surge tank increase the difficulty of optimization of hydraulic parameters.4][15] The SLHT and its coupling with surge tank make the transient processes of HSSLHT significantly different from those with short headrace tunnel.Therefore, the optimization of hydraulic parameters for pipeline system of HSSLHT cannot indiscriminately use the results and methods of that with short headrace tunnel.And for HSSLHT, the contradiction among the optimization requirements of hydraulic parameters for pipelines and surge tank is salient.
The optimization of hydraulic parameters for pipeline system is based on the transient processes of HSSLHT, which are determined by the hydraulic and mechanical coupling.However, the study on the optimization of hydraulic parameters for pipeline system of HSSLHT is still in blank.The related research mainly concern two issues.One issue is the transient processes of hydropower station with long headrace tunnel (HSLHT).The other issue is the optimization of guide vane rules and governor parameters for hydropower station.The literature review is presented as follows.
7][18][19][20][21][22][23] Specifically, Zhang et al. 16 investigate the water hammer characteristics of HSLHT under the effect of air cushion surge tank.The applicability of surge tank to the HSLHT is illuminated.Chen et al. 17 study the design of surge tanks for HSLHT.Different types of surge tanks are compared from the perspective of water hammer and surge wave characteristics.Guo et al. 18 analyze the surge wave superposition characteristics of HSLHT.The optimal superposition strategy of the surge waves in surge tank is proposed to improve the dynamic quality.Wang and Guo 19 investigate energy transfer process in the surge tank and pipelines of HSSLHT.From the point of energy transfer, the stability and hydraulic coupling of transient process of surge wave are revealed.Guo et al. 20 study the surge wave superposition characteristics in series surge tanks of HSSLHT.The applicability and design suggestions of series surge tanks for the HSSLHT are illuminated.Guo and Qu 21 analyze the stability of regulating system of HSSLHT.The method for the description of system stability is put forward.Qu and Guo 22 study the control of surge wave for HSSLHT.The robust control is applied for the strategy design.It is verified that the designed robust strategy can control the transient process of HSSLHT effectively.Qu and Guo 23 investigate the regulating characteristics for primary frequency control of HSSLHT.The dynamic behavior of primary frequency control is evaluated.The influence of pipeline coupling on the regulating characteristics is revealed.
5][26][27][28][29][30][31][32][33] Specifically, Shu et al. 24 introduced the genetic algorithm into the optimization of guide vane closing rule for pumped storage unit.By using the genetic algorithm, the conditions under the highest head and lowest head can be optimized simultaneously.Zhou et al. 25 study the optimization for the broken line closing rule of guide vane.The simulated annealing algorithm is applied, and both the water hammer pressure and rotational speed are considered.It is verified that the obtained closing rule is reasonable.Li et al. 26 studied the joint closing rule of guide vane and ball valve for unit under successive load rejection.The optimization is based on the multiobjective algorithm.It is pointed out that the obtained joint closing rule can coordinate the contradiction between rotational speed and pressure.Hou et al. 27 optimize the transient process of successive starting up of two units.Different types of start-up strategies are compared, and the optimal parameters of governor are given.Fang and Shen 28 study the optimization of governor parameters for hydropower stations.The particle swarm algorithm is adopted.It is verified that the optimized governor parameters can improve the performance of transient process.Lansberry and Wozniak 29 and Meng et al. 30 investigate the parameter tuning of governor for the robustness issue.The genetic algorithm is used.The optimal parameters are obtained and can adapt different conditions.Tang et al. 31 analyze the optimization of turbine governor parameters based on the evolutionary algorithm.That method can obtain the satisfied governor parameters rapidly.Ek Fälth et al. 32 develop a series of hydropower optimization models for a single river with various levels of techno-physical detail to evaluate options for hydropower representations in energy system investment models.It is pointed out that the physical constraints on hydropower affect optimal power production significantly.Nedaei and Walsh 33 propose a novel analytical approach to optimize an efficient hydropower power plant.The optimum energy production and the sensitivity analysis of cost are estimated.
This paper studies the optimization of hydraulic parameters for pipeline system of HSSLHT considering operational scenarios.Because the research topic is an issue of multivariable and multiobjective, the mayfly algorithm is applied for the rapid optimizing.The novelties of the study are (1) Establish the model for the HSSLHT under the effect of load disturbance.(2) Propose the optimization design of hydraulic parameters for pipeline system based on mayfly algorithm, and realize the optimization of hydraulic parameters for pipeline system.(3) Reveal the effects of pipeline diameters and transfer coefficients of turbines (TCT) on the optimization of hydraulic parameters of pipeline systems.
In Section 2, the state equation of HSSLHT under the effect of load disturbance is derived.The optimization design of hydraulic parameters for pipeline system based on mayfly algorithm is proposed.In Section 3, the optimization of hydraulic parameters for pipeline system is conducted and analyzed by simulation.In Sections 4 and 5, the effects of pipeline diameters and TCT on the optimization of hydraulic parameters of pipeline system are revealed.

| MATHEMATICAL FORMULATION
Figure 1 shows the HSSLHT, which contains pipeline system and turbine unit system.The basic thought of this study is the determination of the optimal combination of hydraulic parameters for pipeline system based on mayfly algorithm considering operational scenarios to realize the optimal dynamic performance of HSSLHT.The operational scenarios in the paper refer to the load disturbance during the operation of HSSLHT, which includes load decrease condition and load increase condition.
z q q q q q q z q e x e y q q q q q e x e y e x e y m e x q e x e y e x e y m e x K x Equation ( 7) describes the dynamic response and performance of HSSLHT under load disturbance m g .

| Optimization design of hydraulic parameters for pipeline system based on mayfly algorithm
In the present study, the mayfly algorithm [44][45][46][47] is applied for the optimization of hydraulic parameters for pipeline system.According to the model of HSSLHT, the optimization design of mayfly algorithm of hydraulic parameters for pipeline system is conducted in this section.Firstly, the three elements, that is, decision variables, objective functions and constraint conditions, are determined.Then, the operators of mayfly algorithm are illuminated.Finally, the implementation procedure for the optimization of hydraulic parameters for pipeline system based on mayfly algorithm is designed.

| Decision variables, objective functions, and constraint conditions (1) Decision variables
The decision variables are the hydraulic parameters, which are needed to optimize in the study.The pipeline system contains the SLHT, throttled surge tank, and penstock.For the SLHT, there are three fundamental hydraulic parameters, that is, L H , f H , and h H0 .In practical engineering, f H is determined by the Q 0 and economic velocity.Therefore, L H and h H0 are selected for the optimization.Similarly, L P and h P0 are selected for the optimization.For the throttled surge tank, there are two fundamental hydraulic parameters, that is, F and α S .Those two parameters are selected for the optimization.Moreover, the working head of hydropower station, that is, H 0 , is an important parameter for the design and operation of pipeline system.Therefore, H 0 is also selected as a hydraulic parameter for the optimization.
To sum up, the seven hydraulic parameters, that is, L H , h H0 , L P , h P0 , F , α S , and H 0 , are the decision variables.
(2) Objective functions For the HSSLHT, the dynamic performance is mainly determined by the dynamic response process of turbine unit frequency (DRPTUF) and water level oscillation of throttled surge tank (WLOTST).The performance indexes of the DRPTUF and WLOTST evaluate the performance of HSSLHT.Moreover, those performance indexes are the functions of decision variables and then are selected as the objective functions.
For the DRPTUF, the overshoot σ p1 and settling time t s are the two most important performance indexes about the dynamic performance.Therefore, σ p1 and t s are selected as the objective functions of the DRPTUF.For the WLOTST, the overshoot σ p2 and attenuation rate δ are the two most important performance indexes about the dynamic performance.Therefore, σ p2 and δ are selected as the objective functions of the WLOTST.The above four objective functions, that is, σ p1 , t s , σ p2 , and δ, are the functions of the decision variables., ,  , , , ) = , (3) Constraint conditions The constraint conditions determine the ranges of decision variables.For the HSSLHT, the value ranges of decision variables are codetermined by the design specifications and engineering experience.For different engineering projects, the constraint conditions for the decision variables, that is, L H , h H0 , L P , h P0 , F , α S , and H 0 , are different.The specific ranges of decision variables are presented in the following section of case study based on an engineering project.

| Operators of mayfly algorithm
For a n-dimensional optimization problem, the values of decision variables are regarded as the position of mayfly in the space of decision variables and denoted as u u u u = ( , , , ) . According to the values of decision variables, the value of objective function f u ( ) can be calculated and then used to evaluate the position of mayfly.The movement velocity of mayfly is Because the objective functions are σ p1 , t s , σ p2 , and δ, the optimization of hydraulic parameters for pipeline system is a multiobjective problem.Then, the Pareto optimal solution and congestion degree are introduced in the optimization process.[46][47] (1) Movement of male mayfly The male mayfly adjusts its movement velocity based on the positions of itself and the others.If the position of male mayfly is superior to the optimal individual position of population, the male mayfly presents the random movement, that is, nuptial flight.If the position of male mayfly is inferior to the optimal individual position of population, the male mayfly learns from the optimal individual and changes its movement velocity.Based on the above principles, the updating formulas of the position and movement velocity of male mayfly are presented as follows: (2) Movement of female mayfly The nondominated sort for male and female mayflies yields the nondominated class of mayflies.Based on the calculation of congestion degree for the mayflies with the same class, the sort for the superiority and inferiority of mayflies can be determined.A higher nondominated class and a greater congestion degree indicate a superior mayfly.The female mayfly is attracted by the male mayfly with the same degree of superiority and inferiority, and then mates with the male mayfly.If the position of male mayfly is inferior to the position of female mayfly, the female mayfly presents the random movement.Based on the above principles, the updating formulas of the position and movement velocity of female mayfly are presented as follows: (3) Copulation of mayflies By selecting the mayflies from the Pareto solution set of mayfly algorithm randomly, the copulation of the selected mayflies produces one male offspring and one female offspring.The formulas for the copulation of mayflies are presented as follows: (4) Variation of mayfly If the rate of convergence of mayfly algorithm is too great, the search would converge to the local Pareto front.Therefore, the operator of variation is needed to introduce to make the mayfly jump out of the local Pareto front.The operator of variation conducts the genetic variation for the offspring of mayfly.The formula for the variation of mayfly is presented as follows:

| Implementation procedure for optimization
According to the above analysis, the implementation procedure for the optimization is shown in Figure 2.
Based on Figure 2, the specific implementation procedure is illuminated as follows.
Step 1: The initialization of mayfly population yields the random position of mayfly, that is, the values of decision variables.By substituting the values of decision variables into objective functions yields the values of σ p1 , t s , σ p2 , and δ.Based on the optimiza- tion objective, the nondominated sort for the decision variables is determined, and the congestion degree is calculated.The obtained first-class Pareto nondominated solution is saved in the Pareto optimal solution set (POSS).
Step 2: The global optimal position of mayfly population is determined by selecting the decision variables from the POSS randomly.The position and movement velocity of mayfly are updated according to the updating formulas.
Step 3: The copulation of mayflies in the POSS produces the female offspring mayfly and male offspring mayfly.The operator of variation is conducted for the offspring mayflies.The values of optimization objective are calculated based on the decision variables expressed by the positions of offspring mayflies.By using the values of optimization objective, the nondominated sort for the male and female mayflies including the offspring mayflies is determined, and the congestion degree is calculated.
Step 4: According to the descending orders for the Pareto nondominated class and congestion degree, the male and female mayflies are sorted.Based on the sort results, the mayfly individuals that exceed the maximum of population are removed.
The obtained mayflies and the individuals saved in the POSS are sorted together.Then, the obtained first-class Pareto nondominated solution is saved in the POSS.
Step 5: The cyclic iteration returns to Step 2 and ends until the number of iterations reaches the upper limit.Then, the final POSS is obtained.The final POSS is the parameter set of decision variables that can make the σ p1 , t s , σ p2 , and δ reach the Pareto front.
Based on the model of HSSLHT and optimization design, the optimization of hydraulic parameters for pipeline system of HSSLHT can be carried out based on the mayfly algorithm.The algorithm programming from the math phase to numerical results is realized by MATLAB.Firstly, the model of HSSLHT is programmed, and the ordinary differential equation shown in Equation ( 7) is solved by the ODE45 function in MATLAB.Then, the basic processes of the mayfly algorithm are written into MATLAB codes, including the operators of mayfly algorithm, Pareto nondominated sorting, and congestion degree calculation.Finally, the objective functions of the mayfly algorithm are set as the evaluation indexes for the DRPTUF and WLOTST solved from Equation ( 7) by the ODE45 function.The pipeline system parameters of HSSLHT are calculated as the mayfly parameters.
The basic steps of the mayfly algorithm can be summarized as the following pseudocode:

| SIMULATION ANALYSIS FOR OPTIMIZATION OF HYDRAULIC PARAMETERS FOR PIPELINE SYSTEM
In this section, the optimization results are obtained and the simulation analysis is conducted.For the analysis, a practical HSSLHT is chosen as example.The data of example are presented as follows: H 0 = 288 m, Q 0 = 228.6 m 3 /s, L H = 16,662.16m, f H = 113.1 m 2 , T wH0 = 23.84 s, h H0 = 23.04 m, L P = 530.69m, f P = 34.18m 2 , T wP0 = 1.26 s, h P0 = 5.76 m, F = 411.84m 2 , α S = 0.0004 s 2 /m 5 , e h = 1.5, e y = 1, e x = −1, e qh = 0.5, e qy = 1, e qx = 0, e g = 0, T a = 9.46 s, K p = 1.5, The initial population quantities of male and female mayflies both are 20.The maximum of the POSS is 40.The quantity of offspring mayfly is 20.The variation probability is 0.02.The maximum of the evolution generation is 500.The constraint conditions of decision variables are presented in Table 1.
The initial population of mayfly is produced.The objective functions are calculated and sorted.By using the operators of movement, copulation, and variation, the new populations of mayfly are produced.Based on the sort of populations of mayfly, the nondominated individuals are saved in the POSS.By removing the intersectional individuals, the optimization operations of mayfly continue.If the end condition is satisfied, the final POSS is output.
There are four objective functions, that is, σ p1 , t s , σ p2 , and δ.To illustrate the effect of the weights of objective functions on the optimization results, three different combinations of weighting factors for the objective functions are taken.The first group of weighting factors for σ p1 , t s , σ p2 , and δ are 0.5, 0.5, 0.5, and 0.5, respectively.The second group of weighting factors for σ p1 , t s , σ p2 , and δ are 0.3, 0.7, 0.4, and 0.6, respectively.The third group of weighting factors for σ p1 , t s , σ p2 , and δ are 0.9, 0.1, 0.9, and 0.1, respectively.By using the weighting factors, the four objectives can be converted into single objective.Based on the mayfly algorithm, the single objective under three combinations of weighting factors is calculated and shown in Figure 3.The optimization results of decision variables are determined and shown in Table 2.
Figure 3 and Table 2 show that, under three combinations of weighting factors, the optimization results of decision variables are the same.Therefore, the several objective functions exhibit significant conflicts.Moreover, the minimum evolution generation for optimization changes with the change of weighting factors.In general, the optimization searching speed is rapid, and the minimum evolution generation for optimization is much less than the maximum of the evolution generation.Hence, in the present study, the four objective functions do not have to be converted into single objective function.
To justify the choice of mayfly algorithm, two frequently used algorithms, that is, genetic algorithm [29][30][31] and firefly algorithm, 48,49 are selected for contrastive analysis.The three algorithms are applied to optimize the hydraulic parameters for pipeline system of HSSLHT.The above example of HSSLHT is chosen for calculation.The basic parameters for the three algorithms are shown in Table 3.
By using the mayfly algorithm, genetic algorithm, and firefly algorithm, the Pareto optimal fronts can be obtained and shown in Figure 4.Among the obtained Pareto optimal fronts, the best objective functions are shown in Table 4.
Figure 4 and Table 4 show that, compared to the genetic algorithm and firefly algorithm, the Pareto optimal fronts by using the mayfly algorithm are closer to the ideal Pareto fronts.The best objective functions under the mayfly algorithm are superior to those under the genetic algorithm and firefly algorithm.Compared to the firefly algorithm, the four objective functions σ p1 , t s , σ p2 , and δ under mayfly algorithm are improved by 3.2%, 3.5%, 14.7%, and 0.7%.Compared to the genetic algorithm, the four objective functions σ p1 , t s , σ p2 , and δ under mayfly algorithm are improved by 0.3%, 2.0%, 5.1%, and 0.2%.
Moreover, the optimization process of mayfly algorithm needs a shorter calculation time and has a faster convergence rate.The optimization of hydraulic parameters for pipeline system of HSSLHT needs a large amount of computation.Therefore, the mayfly algorithm is suitable for the specific optimization task.The mayfly algorithm can avoid trapping in local optimum.
To sum up, the mayfly algorithm outweighs the other algorithms for the specific configuration, that is, optimization of hydraulic parameters for pipeline system of HSSLHT.
The multiobjective optimization results are sorted by the Pareto nondominated sorting method.In this section, the load decrease condition and load increase condition are selected for the simulation analysis to illustrate the optimization processes and results of hydraulic parameters for pipeline system.

| Optimization under load decrease condition
For the load decrease condition, we take m g = −0.1.By using the optimization method of hydraulic parameters for pipeline system based on mayfly algorithm, the Pareto optimal fronts can be obtained and shown in Figure 5.Among the obtained Pareto optimal fronts, eight groups of representative fronts are selected.For those eight groups of representative fronts, the values of decision variables and objective functions are shown in Table 5.Under the eight groups of decision variables, the DRPTUF and WLOTST are shown in Figure 6.
Figures 5 and 6 and Table 5 show that T A B L E 2 Optimization results of decision variables under three combinations of weighting factors.

Combinations of weighting factors
Minimum evolution generation 0.5, 0.5, 0. (1) By using the multiobjective mayfly algorithm, the optimization method can make the hydraulic parameters for pipeline system reach the POSS under the load decrease condition.Meanwhile, the objective functions approach the Pareto optimal fronts gradually.Among the four objective functions, the rapidity and stability of the DRPTUF are contradictory.When σ p1 becomes less, t s becomes greater.Similarly, the rapidity and stability of the WLOTST are contradictory.When σ p2 becomes less, δ becomes less.(2) Among the obtained eight groups of decision variables, the values of H 0 , F , and α S approach their upper limits, the values of L H and h P0 approach their lower limits, while the values of h H0 and L P distribute uniformly between their lower limits and upper limits.Therefore, the H 0 , F , and α S can coordinate the different objective functions, and the great values of H 0 , F , and α S are favorable for the dynamic performance of HSSLHT.The L H and h P0 also can coordinate the different objective functions, and the small values of L H and h P0 are favorable for the dynamic performance of HSSLHT.The h H0 and L P cannot coordinate the different objective functions, and the certain values of h H0 and L P are only favorable for the part indexes of the dynamic performance of HSSLHT.

| Optimization under load increase condition
For the load increase condition, we take m g = 0.1.By using the optimization method of hydraulic parameters for pipeline system based on mayfly algorithm, the Pareto optimal fronts can be obtained and shown in Figure 7.Among the obtained Pareto optimal fronts, eight groups of representative fronts are selected.For those eight groups of representative fronts, the values of decision variables and objective functions are shown in Table 6.Under the eight groups of decision variables, the DRPTUF and WLOTST are shown in Figure 8. Figures 7 and 8 and Table 6 show that, under the load increase condition, the optimization method can make the hydraulic parameters for pipeline system reach the POSS, and the objective functions approach the Pareto optimal fronts gradually.When σ p1 becomes less, t s becomes greater.Similarly, when σ p2 becomes less, δ becomes less.Among the obtained eight groups of decision variables, the values of H 0 and F approach their upper limits, the values of L H , h P0 , and α S approach their lower limits, while the values of h H0 and L P distribute uniformly between their lower limits and upper limits.Therefore, the H 0 and F can coordinate the different objective functions, and the great values of H 0 and F are favorable for the dynamic performance of HSSLHT.The L H , h P0 , and α S also can coordinate the different objective functions, and the small values of L H , h P0 , and α S are favorable for the dynamic performance of HSSLHT.The h H0 and L P cannot coordinate the different objective functions, and the certain values of h H0 and L P are only favorable for the part indexes of the dynamic performance of HSSLHT.For the pipeline system, the pipeline diameters are important hydraulic parameters.In the optimization procedure, the pipeline diameters are not selected as decision variables.The effect of pipeline diameters on the optimization of hydraulic parameters of pipeline system is analyzed.The pipeline diameters include the diamater of SLHT D H and diameter of penstock D P .Figure 9 shows that (1) D H and D P have an obvious effect on the Pareto optimal fronts of σ p1 , t s , σ p2 , and δ.With the increase of D H , both the fronts of σ p1 and t s firstly increase and then decrease.Accordingly, the performance for the DRPTUF firstly becomes worse and then becomes better.With the increase of D H , the front of σ p2 decreases, while the front of δ increases.

Groups of fronts H
Accordingly, the performance for the WLOTST becomes better.(2) With the increase of D P , the front of σ p1 decreases, while the front of t s increases.Accordingly, the performance for the DRPTUF does not simply become worse or better.With the increase of D P , the front of σ p2 decreases, while the front of δ increases.Accordingly, the performance for the WLOTST becomes better.By comparison, the effect   of D H on the fronts of σ p1 , t s , σ p2 , and δ is more significant than that of D P on the fronts of σ p1 , t s , σ p2 , and δ.The effects of D H and D P on the fronts of σ p2 and δ are more significant than those of D H and D P on the fronts of σ p1 and t s .

Groups of fronts H
For all the obtained Pareto optimal fronts under different values of D H and D P in Figure 9, the corresponding values of decision variables can be determined.Under the determined decision variables, the HSSLHT is certain.Because there are many groups of Pareto optimal fronts, the HSSLHT has many combinations of hydraulic parameters that can satisfy the dynamic performance requirements.To evaluate the dynamic performance of HSSLHT under different combinations of hydraulic parameters, the robustness index (RI) 50,51 is adopted.Under different values of D H and D P , the values of RI for all the Pareto optimal fronts are determined and shown in Figures 10A and 11A, respectively.Then, based on Figures 9, 10a, and 11A, the representative fronts under different values of D H and D P are selected among the obtained Pareto optimal fronts.For the selected representative fronts, the values of decision variables and objective functions are shown in Table 7.Under the decision variables, the DRPTUF and WLOTST are shown in Figures 10B,C and 11B,C.
Figures 10 and 11 and Table 7   change of D P , the optimization results for the decision variables of L P and F have significant changes, while the optimization results for the other decision variables only have slight changes.Therefore, the optimization results of L P and F are the most sensitive for the changes of D H and D P .L P mainly affects the DRPTUF, and F mainly affects the WLOTST.L P and F are the two key hydraulic parameters for the optimization of pipeline system.

| EFFECT OF TCT ON OPTIMIZATION OF HYDRAULIC PARAMETERS
The mechanical components of HSSLHT have effect on the design of pipeline system.Moreover, the mechanical components are mainly related with the operation of HSSLHT.The pipeline system should match the operation of HSSLHT.Among the mechanical components of   HSSLHT, the turbine couples with the pipeline system directly.In this section, the effect of TCT on the optimization of hydraulic parameters of pipeline system is analyzed.The TCT includes the discharge transfer coefficients, that is, e qh , e qx , and e qy , and the moment transfer coefficients, that is, e h , e x , and e y .

| Effect of discharge TCT on optimization
The effect of discharge TCT on the optimization of hydraulic parameters is analyzed in this section.e qh , e qx , and e qy are taken as different values.For the e qh , the values of 0.3, 0.4, 0.6, and 0.7 are taken, respectively.For the e qx , the values of −0.2, −0.1, 0.1, and 0.2 are taken, respectively.For the e qy , the values of 0.8, 0.9, 1.1, and 1.2 are taken, respectively.Under different values of e qh , e qx , and e qy , the Pareto optimal fronts are obtained and shown in Figure 12.
Figure 12 shows that (1) e qh has a slight effect on the Pareto optimal fronts of σ p1 and t s , while has an obvious effect on the Pareto optimal fronts of σ p2 and δ.With the increase of e qh , the fronts of σ p1 and t s change slightly without obvious rule.The front of σ p2 firstly increases and then decreases, while the front of δ firstly decreases and then increases.Accordingly, the performance for the WLOTST firstly becomes worse and then becomes better.(2) e qx has an obvious effect on the Pareto optimal fronts of σ p1 and t s , while has a slight effect on the Pareto optimal fronts of σ p2 and δ.With the increase of e qx , both the fronts of σ p1 and t s decrease.Accordingly, the performance for the DRPTUF becomes better.With the increase of e qx , the fronts of σ p2 and δ change slightly without obvious rule.(3) e qy has an obvious effect on the Pareto optimal fronts of σ p1 , t s , σ p2 , and δ.With the increase of e qy , the front of σ p1 increases, while the front of t s decreases slightly.The increase of σ p1 plays a dominant role.Accordingly, the performance for the DRPTUF becomes worse.With the increase of e qy , the front of σ p2 increases, while the front of δ decreases.Accordingly, the performance for the WLOTST becomes worse.
Under different values of e qh , e qx , and e qy , the values of RI for all the Pareto optimal fronts are determined and shown in Figures 13A, 14A, and 15A, respectively.Then, based on Figures 12, 13A, 14A, and 15A, the representative fronts under different values of e qh , e qx , and e qy are selected among the obtained Pareto optimal fronts.For the selected representative fronts, the values of decision variables and objective functions are shown in Table 8.Under the decision variables, the DRPTUF and WLOTST are shown in Figures 13B,C, 14B,C, and 15B,C.
Figure 13-15 and Table 8 show that (1) For the values of 0.3, 0.4, 0.6, and 0.7 of e qh , the mean values of the RI for all the Pareto optimal fronts are 0.3883, 0.2935, 0.2006, and 0.1664, respectively.For the values of −0.2, −0.1, 0.1, and 0.2 of e qx , the mean values of the RI for all the Pareto optimal fronts are 0.2383, 0.2509, 0.2532, and 0.2434, respectively.For the values of 0.8, 0.9, 1.1, and 1.2 of e qy , the mean values of the RI for all the Pareto optimal fronts are 0.2312, 0.2364, 0.2496, and 0.2500, respectively.Therefore, e qh has an obvious effect on the robustness of dynamic performance of HSSLHT, while e qx and e qy have a slight effect on the robustness of dynamic performance of HSSLHT.With the increase of e qh , the robustness becomes better, and the capacity of resisting disturbance of HSSLHT becomes better.When e qh increases by 233%, the value of RI of HSSLHT is improved by 43%.When e qy increases by 50%, the value of RI of HSSLHT deteriorates by 8%.(2) With the change of e qh , the optimization results for the decision variables of L H , L P , and α S have significant changes, while the optimization results for the other decision variables only have slight changes.With the change of e qx , the optimization results for the decision variable of h H0 have signifi- cant change, while the optimization results for the other decision variables only have slight changes.With the change of e qy , the optimization results for the decision variable of h H0 , L P , and α S have significant changes, while the optimization results for the other decision variables only have slight changes.Therefore, the optimization results of h H0 , L P , and α S are the most sensitive for the changes of e qh , e qx , and e qy .h H0 , L P , and α S are the three key hydraulic parameters for matching the operation of HSSLHT from the perspective of discharge TCT.

| Effect of moment TCT on optimization
The effect of moment TCT on the optimization of hydraulic parameters is analyzed in this section.e h , e x , and e y are taken as different values.For the e h , the values of 1.3, 1.4, 1.6, and 1.7 are taken, respectively.For the e x , the values of −0.8, −0.9, −1.1, and −1.2 are taken, respectively.For the e y , the values of 0.8, 0.9, 1.1, and 1.2 are taken, respectively.Under different values of e h , e x , and e y , the Pareto optimal fronts are obtained and shown in Figure 16.
Figure 16 shows that (1) e h , e x , and e y have an obvious effect on the Pareto optimal fronts of σ p1 , t s , σ p2 , and δ.With the increase of e h , the front of σ p1 increases, while the front of t s decreases.Accordingly, the performance for the DRPTUF does not simply become worse or better.
With the increase of e h , the front of σ p2 increases, while the front of δ decreases.Accordingly, the performance for the WLOTST becomes worse.(2) With the increase of e x , the front of σ p1 increases, while the front of t s decreases.Accordingly, the performance for the DRPTUF does not simply become worse or better.With the increase of e x , the front of σ p2 firstly decreases and then increases, while the front of δ firstly increases and then decreases.Accordingly, the performance for the WLOTST firstly becomes better and then becomes worse.(3) With the increase of e y , both the fronts of σ p1 and t s decrease.Accordingly, the performance for the DRPTUF becomes better.With the increase of e y , the front of σ p2 decreases, while the front of δ increases.Accordingly, the performance for the WLOTST becomes better.
Under different values of e h , e x , and e y , the values of RI for all the Pareto optimal fronts are determined and shown in Figures 17A, 18A

| CONCLUSIONS
The state equation of HSSLHT under the effect of load disturbance is derived.The optimization design of hydraulic parameters for pipeline system based on mayfly algorithm is proposed.The optimization of hydraulic parameters is conducted and analyzed.The effects of pipeline diameters and TCT on the optimization of hydraulic parameters of pipeline system are revealed.The conclusions are: (1) The optimization of hydraulic parameters for pipeline system is a multiobjective problem.For the optimization design of hydraulic parameters for pipeline system based on mayfly algorithm, the seven hydraulic parameters, that is, L H , h H0 , L P , h P0 , F , α S , and H 0 , are the decision variables.σ p1 and t s are the objective functions of the DRPTUF, and σ p2 and δ are the objective functions of the WLOTST.The four objective functions are the functions of the decision variables.The several objective functions exhibit significant conflicts.Compared to the firefly algorithm, the four objective functions σ p1 , t s , σ p2 , and δ under mayfly algorithm are improved by 3.2%, 3.5%, 14.7%, and 0.7%.Compared to the genetic algorithm, the four objective functions σ p1 , t s , σ p2 , and δ under mayfly algorithm are improved by 0.3%, 2.0%, 5.1%, and 0.2%.(2) The optimization method by using the multiobjective mayfly algorithm can make the hydraulic (3) The effect of D H on the fronts of σ p1 , t s , σ p2 , and δ is more significant than that of D P on the fronts of σ p1 , t s , σ p2 , and δ.The effects of D H and D P on the fronts of σ p2 and δ are more significant than those of D H and D P on the fronts of σ p1 and t s .D P has an obvious effect on the robustness of dynamic performance of HSSLHT.With the increase of D P , the robustness of HSSLHT becomes worse.The optimization results of L P and F are the most sensitive for the changes of D H and D P .L P and F are the key hydraulic parameters for the optimization of pipeline system.When D P increases by 30%, the value of RI of HSSLHT deteriorates by 57%.HSSLHT deteriorates by 8%.When e y increases by 50%, the value of RI of HSSLHT deteriorates by 39%.
The following research prospects and directions will be focused in the future.
(1) The HSSLHT is a coupling system of pipeline system and turbine unit system.In the present study, the hydraulic parameters for pipeline system are optimized based on the mayfly algorithm.During the optimization process, the mechanical and electrical parameters for turbine unit system are given.The dynamic performance of HSSLHT is affected by the combined effect of pipeline system and turbine unit system.Moreover, the optimization results of hydraulic parameters for pipeline system are influenced by the mechanical and electrical parameters for turbine unit system.Therefore, the joint optimization of the hydraulic parameters for pipeline system and the mechanical and electrical parameters for turbine unit system is more reasonable and worth studying.(2) The object of the present study is the HSSLHT.At present, the pumped storage power station plays an important role in the power grid.More and more pumped storage power stations are developed and constructed.The optimization method proposed in the paper can be potentially applied to determine the optimal parameters for pumped storage power station.That is a meaningful research topic.When the optimization object is changed to pumped storage power station, the features of pumped storage power station need to be considered during the optimization design.
The pumped storage power station has complex layouts of pipeline system, such as the cyclic pipeline system with multiple units, and the upstream and downstream double surge tanks.Moreover, the pumped storage power station has both the generating condition and pumping condition.

F I G U R E 2
Implementation procedure for optimization.
and velocities v mi Initialize the female mayfly population u fi (i = 1, 2,…, M) and velocities v fi Solving the HSSLHT differential equation and evaluating solutions Store the nondominated solutions found in an external repository Sort the mayflies based on congestion level Do while stopping criteria are not met Update velocities and solutions of males and females (Continues) Solving the HSSLHT differential equation and evaluating solutions If a new mayfly dominates its personal best Replace personal best with the new solution If no one dominates the other The new solution has a chance of 50% to replace the personal best Rank the mayflies Mate the mayflies Evaluate offspring Separate offspring to male and female randomly If an offspring dominates its same-sex parent Replace parent with the offspring Insert all the new nondominated solutions found in the external repository Sort the nondominated solutions and truncate the repository if needed End while stopping criteria are met Postprocess results and visualization of nondominated solutions T A B L E 1 Constraint conditions of decision variables.

F I G U R E 4
Pareto optimal fronts by using the mayfly algorithm, genetic algorithm and firefly algorithm.(A) Fronts of σ p1 and t s .(B) Fronts of σ p2 and δ.T A B L E 4The best objective functions under mayfly algorithm, genetic algorithm, and firefly algorithm.
D H and D P are taken as different values.For the D H , the values of 10, 11, 13, and 13.6 m are taken, respectively.For the D P , the values of 5.4, 6, 6.4, and 7 m are taken, respectively.The load condition is taken as m g = −0.1.Under different values of D H and D P , the Pareto optimal fronts are obtained and shown in Figure 9.

F I G U R E 5
Pareto optimal fronts under load decrease condition m g = −0.1.(A) Fronts of σ p1 , t s , σ p2 and δ. (B) Fronts of σ p1 and t s .(C) Fronts of σ p2 and δ.T A B L E 5 Values of decision variables and objective functions for the eight groups of representative fronts under m g = −0.1.

F I G U R E 6
DRPTUF and WLOTST under the eight groups of decision variables under m g = −0.1.(A) DRPTUF.(B) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; WLOTST, water level oscillation of throttled surge tank.

T A B L E 6
Values of decision variables and objective functions for the eight groups of representative fronts under m g = 0.1.

F I G U R E 8
DRPTUF and WLOTST under the eight groups of decision variables under m g = 0.1.(A) DRPTUF.(B) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; WLOTST, water level oscillation of throttled surge tank.F I G U R E 7 Pareto optimal fronts under load decrease condition m g = 0.1.(A) Fronts of σ p1 , t s , σ p2 and δ. (B) Fronts of σ p1 and t s .(C) Fronts of σ p2 and δ.

F I G U R E 10
Values of RI for all the Pareto optimal fronts and the DRPTUF and WLOTST for the representative fronts under different values of D H . (a) Values of RI.(b) DRPTUF.(c) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.T A B L E 7 Values of decision variables and objective functions for the representative fronts under different values of D H and D P .Parameters Values (m) H 0 (m) L H (m) h H0 (m) L P (m) h P0 (m) F (m 2 ) α S (s 2 /m

F I G U R E 11
Values of RI for all the Pareto optimal fronts and the DRPTUF and WLOTST for the representative fronts under different values of D P .(A) Values of RI. (B) DRPTUF.(C) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.
, and 19A, respectively.Then, based on Figures16, 17A, 18A, and 19A, the representative fronts under different values of e h , e x , and e y are selected among the obtained Pareto optimal fronts.For

F
I G U R E 13 Values of RI for all the Pareto optimal fronts and the DRPTUF and WLOTST for the representative fronts under different values of e qh .(A) Values of RI. (B) DRPTUF.(C) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.F I G U R E 14 Values of RI for all the Pareto optimal fronts and the DRPTUF and WLOTST for the representative fronts under different values of e qx .(A) Values of RI. (B) DRPTUF.(C) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.F I G U R E 15 Values of RI for all the Pareto optimal fronts and the DRPTUF and WLOTST for the representative fronts under different values of e qy .(A) Values of RI. (B) DRPTUF.(C) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.the selected representative fronts, the values of decision variables and objective functions are shown in

F I G U R E 16
Pareto optimal fronts under different e h , e x , and e y .(A) Pareto optimal fronts under different e h .(a-1) Fronts of σ p1 , t s , σ p2 , and δ. (a-2) Fronts of σ p1 and t s .(a-3) Fronts of σ p2 and δ. (B) Pareto optimal fronts under different e x .(b-1) Fronts of σ p1 , t s , σ p2 , and δ. (b-2) Fronts of σ p1 and t s .(b-3) Fronts of σ p2 and δ. (C) Pareto optimal fronts under different e y .(c-1) Fronts of σ p1 , t s , σ p2 , and δ. (c-2) Fronts of σ p1 and t s .(c-3) Fronts of σ p2 and δ.(4) e qh and e y have an obvious effect on the robustness of dynamic performance of HSSLHT.With the increase of e qh and e y , the robustness of HSSLHT becomes better.The optimization results of h H0 , L P , and α S are the most sensitive for the changes of e qh , e qx , and e qy .The optimization results of L H , L P , h P0 , and α S are the most sensitive for the changes of e h , e x , and e y .L H , L P , h H0 , h P0 , and α S are the key hydraulic parameters for matching the operation of HSSLHT from the perspective of TCT.When e qh increases by 233%, the value of RI of HSSLHT is improved by 43%.When e qy increases by 50%, the value of RI of F I G U R E 17 Values of RI for all the Pareto optimal fronts and the DRPTUF and WLOTST for the representative fronts under different values of e h .(A) Values of RI. (B) DRPTUF.(C) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.F I G U R E 18 Values of RI for all the Pareto optimal fronts and the DRPTUF and WLOTST for the representative fronts under different values of e x .(A) Values of RI. (B) DRPTUF.(C) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.F I G U R E 19 Values of RI for all the Pareto optimal fronts and the DRPTUF and WLOTST for the representative fronts under different values of e y .(A) Values of RI. (B) DRPTUF.(C) WLOTST.DRPTUF, dynamic response process of turbine unit frequency; RI, robustness index; WLOTST, water level oscillation of throttled surge tank.
subscript 0 represents the initial value of variable.
With the change of D H , the optimization results for the decision variables of L P , F , and α S have significant changes, while the optimization results for the other decision variables only have slight changes.With the

Table 9 .
Under the decision variables, the DRPTUF and WLOTST are shown in Figures 17B,C, 18B,C, and 19B,C.With the changes of e h and e x , the optimization results for the decision variables of L H , L P , h P0 , and α S have significant changes, while the optimization results for the other decision variables only have slight changes.With the change of e y , the optimization results for the decision variables of L H , h H0 , L P , and α S have significant changes, while the optimization results for the other decision variables only have slight changes.Therefore, the optimization results of L H , L P , h P0 , and α S are the most sensitive for the changes of e h , e x , and e y .L H , L P , h P0 , and α S are the four key hydraulic parameters for matching the operation of HSSLHT from the perspective of moment TCT.
Values of decision variables and objective functions for the representative fronts under different values of e qh , e qx , and e qy .Parameters Values (m) H 0 (m) L H (m)h H0 (m) L P (m) h P0 (m) F (m 2 ) α S (s 2 /m 5 ) σ p1 (%) t s (s) σ p2 (%) δ (%) When σ p1 and σ p2 become less, t s becomes greater and δ becomes less.H 0 , F , α S , L H , and h P0 can coordinate the different objective functions, while h H0 and L P cannot coordinate the different objec- tive functions.
Values of decision variables and objective functions for the representative fronts under different values of e h , e x , and e y .Parameters Values (m) H 0 (m) L H (m)h H0 (m) L P (m) h P0 (m) F (m 2 ) α S (s 2 /m 5 ) σ p1 (%) t s (s) σ p2 (%) δ (%) N n (0,1) random number of standard normal distribution with average value of 0 and variance of 1 pbest ij jth dimension position corresponding to the optimal position of ith male mayfly