Optimization of geometric parameters of hydraulic turbine runner in turbine mode based on ISMA and BPNN

The hydraulic turbine in turbine mode (TMHT) has significant advantages in residual energy recovery, but rapid optimization of its runner parameters has always been a challenge. To address this issue, the optimization algorithm ISMA‐BPNN is proposed based on the improved slime mold algorithm (ISMA) and back propagation neural network (BPNN). First, an efficiency characteristic neural network (ECNN) and a water head characteristic neural network (HCNN), which take the geometric parameters of the runner as input, and the efficiency and water head as outputs, respectively, are constructed by combining the orthogonal test‐based sample data, computational fluid dynamics (CFD), and BPNN. Then, the slime mold algorithm is improved and the optimal runner geometric parameters are obtained based on the ISMA, ECNN, and HCNN, to achieve rapid optimization of the TMHT runner. Finally, the CFD‐based numerical calculation accuracy is verified through real machine tests, and the feasibility of the ISMA‐BPNN‐based rapid optimization strategy for TMHT performance is further verified through CFD numerical simulation.


| INTRODUCTION
6][7] Given the advantages of pump reversal as a hydraulic turbine, many scholars have conducted relevant research on it.Carravetta 8 and Mohammad 9 conducted a study on the use of reverse pumps for residual pressure recovery in water supply networks.The research results show that reverse pumps have problems with narrow efficiency zones and low average efficiency when used for energy recovery.On the basis of experimental research, Ye 10 applies numerical simulation to calculate the turbine operating conditions of multi-stage pumps and uses entropy generation theory to study the flow characteristics of multi-stage pump hydraulic turbines under different operating conditions.Furthermore, through relevant analysis, the physical field and energy loss are correlated.The research results show that the multi-stage pump has a higher water head but lower efficiency compared to single-stage pumps.To solve the problems of low efficiency and narrow efficiency zones in the reverse pump type hydraulic turbine, Zhang 11 studied the TMHT, and the results show that the multi-stage TMHT has a wider efficiency zone and higher efficiency at the rated operating point, which has great application prospects.
Due to the significant effect of TMHT on residual energy recovery, its performance optimization has become a research hotspot in the industry.In the research process, to improve the accuracy of simulation calculations, the performance prediction of TMHT is generally combined with computational fluid dynamics (CFD) and Ansys software.However, due to the speed of simulation calculation, the performance improvement of hydraulic turbines cannot be applied with intelligent optimization algorithms.With the increasing demand for TMHT in the industry, traditional performance optimization methods are clearly no longer applicable.Therefore, it is necessary to study the rapid optimization strategy for the TMHT performance.
As the core flow passage component, the geometric parameters of the runner directly affect the TMHT performance.Considering that the working head of TMHT can be flexibly adjusted through the front valve, rapid optimization of runner parameters can serve as a starting point for improving TMHT performance.To improve the TMHT performance, Chen 5 aims to optimize the runner geometric parameters based on the orthogonal test and CFD, with the goals of optimal efficiency and optimal head, respectively.The results show that after optimizing the runner parameters, the efficiency or water head of TMHT is significantly improved.However, the orthogonal test usually assumes a linear relationship between factors and responses, has limited applicability, and has the disadvantage of easily missing optimal solutions.The relationship between the geometric parameters of the runner and the TMHT performance is nonlinear.If it can be expressed reasonably, advanced intelligent optimization algorithms can be further applied to quickly improve the TMHT performance.
6][17] It has the characteristics of a simple structure, fast optimization speed, and can achieve a balance between local optimization and global search.It has good performance in handling nonlinear problems.
Therefore, this paper proposes a fast optimization strategy (ISMA-BPNN) for TMHT performance.First, by combining orthogonal test-based sample data with BPNN, an efficiency characteristic neural network (ECNN) and a water head characteristic neural network (HCNN) are constructed, which take runner geometric parameters as input, and TMHT efficiency and head as output, respectively.Then, based on ISMA, the optimal runner geometric parameters are obtained to achieve rapid optimization of TMHT.Finally, the numerical calculation accuracy based on CFD is verified through real machine tests, and the feasibility of the ISMA-BPNN-based rapid optimization strategy for TMHT performance is further verified through CFD numerical simulation.
To sum up, the main contributions of this study include 1.Based on BPNN, ECNN, and HCNN are constructed, which take runner geometric parameters as input, and TMHT efficiency and head as output respectively.2. SMA is improved and verified through test functions.3. Based on the ISMA and two constructed neural network models, the optimal geometric parameters of the runner are obtained.4.After obtaining the optimal geometric parameters of the runner, the TMHT model is constructed and the ISMA optimization effect is verified based on CFD.
The rest of this article is organized as follows.Section 2 introduces the design method of the TMHT runner in detail.Then, ECNN and HCNN are established through BPNN in Section 3. Subsequently, Section 4 optimizes the SMA and further optimizes the geometric parameters of the TMHT runner.Section 5 verifies the optimization results of the geometric parameters of the TMHT runner.Finally, the conclusions are provided in Section 6.

| THEORETICAL ANALYSIS
The TMHT incorporates various components such as the first stage guide vane, inlet chamber, two runners, guide vane in two stages, and outlet room. 5Among them, the Francis runner is adopted due to its strong ability to recover water heads. 5The design of TMHT is based on the hydraulic turbine design principle and circulation concept. 5The runner design of TMHT relies on two key assumptions: (1) the water enters the runner smoothly without any impact; (2) the circumferential velocity component of the absolute velocity of water at the runner outlet v u2 = 0 m/s.Under these two assumptions, the water entering the runner has sufficient circulation to convert the water head into mechanical energy. 11he fundamental equation of the hydraulic turbine is represented by Equation (1).
where H is the working head, ω is the angular speed of the runner, H e is the effectively utilized head, η is the hydraulic efficiency, g is gravitational acceleration, C 1 and C 2 are the circulations of water at the runner inlet and outlet as shown in Equations ( 2) and (3) (m/s).
where v u1 and v u2 are the circumferential components of the absolute velocity of water at the runner inlet and outlet respectively (m/s), and D 1 and D 2 are the runner diameter at the inlet and outlet, respectively (m).
Under the assumption of normal outlet flow, H e can be mathematically described as Equation ( 4).
Figure 1 shows the inlet and outlet triangles of the runner. 5ccording to Figure 1, H E is transformed as shown in Equation ( 5).
( ) where u 1 is the peripheral speed at the runner inlet; β 1 is the installation angle of the runner inlet; v m1 is the shaft surface speed at the runner inlet; Q is the design discharge; and B is the runner inlet height.
The other flow passage components other than the runner, such as guide vanes, volutes, and outlet chambers, are designed according to the circulation requirements of the hydraulic turbine. 5

| MODELING OF TMHT CHARACTERISTICS BASED ON BPNN
The ECNN and water HCNN take the geometric parameters of the runner (inlet diameter D 1 , inlet width B 1 , inlet placement angle β 1 , outlet diameter D 2 , and number of blades Z) as the inputs and take efficiency and water head as the outputs, respectively, are constructed.Among them, BPNN training data is obtained based on the orthogonal test and CFD under rated operating conditions, as shown in Appendix A. The number of hidden layers of ECNN and HCNN is determined as 10 by the "trial and error method", with root-mean-square deviation RMSE 18 and similarity coefficient R 19 as the evaluation criteria, that is, both network structures are 5-10-1.The structure of ECNN and HCNN is shown in Figure 2. The calculation methods of RMSE and R are shown in Equations ( 6) and (7).The modeling effects of ECNN and HCNN are shown in Figure 3. Furthermore, the RMSE and R calculation results of the two networks are shown in Table 1.From Figure 3, it can be seen that the fitting errors of ECNN and HCNN are both small; the fitting effect of ECNN is better than that of HCNN.From Table 1, it can be seen that the R of both networks is close to 1, and the RMSE is small, indicating that the network fitting effect is excellent.
where d i and y i represent the predicted data and sample data of efficiency or water head respectively, n represents the number of sample data, d ¯and y ¯represent the average value of efficiency and water head in the predicted data and sample data, respectively.
where max(x) represents maximum x, min(x) represents minimum x.  9) and ( 10); NN out represents output of ECNN or HCNN.

| PERFORMANCE OPTIMIZATION OF TMHT BASED ON ISMA
After constructing ECNN and HCNN, rapid performance optimization of TMHT can be achieved based on ISMA.

| Standard SMA
The SMA mainly simulates the foraging behavior and state change of Physarum polycephalum in nature under different food concentrations. 20The slime molds mainly secrete enzymes to digest food, with the front end extending into a fan-shaped shape and the back end surrounded by interconnected venous networks. 17Different concentrations of food in the environment affect the flow of cytoplasm in the venous network of slime molds, resulting in different states of foraging for slime molds. 17hen slime molds approach food, the mathematical model of the SMA can be expressed as follows: where t is the current iteration number, X t ( ) b is the optimal position of the slime mold individual during the tth iteration, X t ( ) m and X t ( ) are the position of two randomly selected slime mold individuals.vb is the control parameter range a a [− , ], vc is the parameter that linearly decreases from 1 to 0, and r is a random number between [0,1].W is the mass of slime mold, representing the weight of fitness.The mathematical model for controlling variable p and parameter vb are where ( ) is the ith slime mold fitness value, F b is the best fitness value among all iterations.The a and W satisfy where t max is the maximum iteration number; "condition" represents the slime mold individuals with fitness values ranking in the top half; "fitness sequence" is the fitness value sequence of slime molds, which uses an ascending arrangement method when solving the minimum value problem; F w represents the worst fitness value among the current iterations.When slime molds wrap food, the mathematical model of the SMA can be expressed as where B u and B l represent the upper and lower boundaries of the search range, respectively, rand and r represent random values between [0, 1], and z is a custom parameter set to 0.03.When slime molds grab food, their venous tissues and biological oscillators undergo changes.The higher the concentration of food in venous contact, the stronger the waves generated by the biological oscillator.Relying on this change, slime molds will grab higher concentrations of food.The changes in the width of the slime vein were achieved using W, vb, and vc.
W simulated the oscillation frequency of slime molds near different food concentrations.vb varies randomly in a a [− , ] and gradually approaches zero as the number of iterations increases.The value of vc oscillates in [−1, 1] and eventually approaches 0. When slime molds choose food, the mutual synergy between vb and vc plays an important role.

| Improved SMA
To avoid the probability of SMA falling into local optima, chaotic mapping is used to initialize the slime mold population; to improve the global search ability of SMA, a weight w that varies with the number of iterations is added to the position update formula during the process of slime mold wrapping food.

Chaotic mapping-based population initialization
The chaotic mapping function Logistic, 21 which has significant randomness and is relatively uniform in [−0.5, 0.5], is adopted and expressed as where A and B are adjustable parameters and are taken as 0.5 and 0.4, respectively.

Nonlinear weight coefficient
The characteristic of w is that the values in the early iteration are larger, while the values in the later iteration are smaller.In the early stage of algorithm iteration, selecting a larger search step size can improve the global search ability, improve the algorithm's global traversal effect, and avoid premature convergence; in the later stage of iteration, selecting a smaller search step can improve the local search and speed up the convergence rate.The calculation of w is shown as The improved position update formula is

| ISMA performance verification
To test the ISMA performance, the SMA, particle swarm optimization (PSO), 22 biogeography algorithm (BBO) 23 , and gravitational search algorithm (GSA) 24 are used to design comparative experiments and solve four typical problems 25 , respectively.The mathematical description of the problem and the three-dimensional surface diagram are shown in Table 2 and Figure 4, respectively.From Figure 4, it can be seen that the sphere function is a simple unimodal function with only one global minimum value; the Griewank function exhibits intense multimodal functions, making it difficult to find its global extremum; the Resinbrock function is unimodal, and its global extremum is difficult to find due to the nonconvex ill-conditioned nature of the interactions between variables; the Rastigin function has multiple local extrema, which makes the algorithm prone to falling into local optima.The calculation process for the optimal parameters of the benchmark test function based on ISMA T A B L E 2 Mathematical descriptions of four benchmark functions.

| Model construction and grid partitioning
Based on the original scheme, the scheme with maximum η (η max scheme), and the scheme with maximum H (H max scheme), the design discharge is obtained to be 0.8155, 0.8706, and 0.9864 m 3 /s, respectively.A two-stage TMHT with volute, first-stage guide vane, interstage guide vane, two runners, and outlet chamber is constructed, as shown in Figure 8.Then, structured grid partitioning is performed on each flow passage component, and the overall computational domain grid is generated through the assembly.Figure 9 is a schematic diagram of the grid model in the calculation domain of the TMHT.For the turbine studied in this article, to save computer resources and accelerate computational convergence, unstructured grids are used for grid partitioning.When conducting numerical calculations on a TMHT, the density of the grid has an impact on the calculation results.Before conducting research on TMHT performance, it is necessary to determine the appropriate number of grids.By setting different grid sizes for the turbine, five grid schemes are ultimately generated.By processing the calculation results of each grid number, the relationships between grid number, η, and H are obtained, as shown in Figure 10.From Figure 10, it can be seen that when the total number of grids is around 3.5 million, the η and H fluctuation are both within 0.3%.Taking into account the numerical calculation time and accuracy, the TMHT model has around 3.5 million grids.

| Comparative analysis
The comparison results between the optimal schemes and the original solution are shown in Figures 11 and 12.It can be seen from Figures 11 and 12 that compared to the original scheme, the η, H, and P of the η max scheme have been improved, with smaller head losses L H under the rated and low discharge conditions; the H max scheme has a slight decrease in η and an increase in L H under large discharge conditions, but the P and the ability to recover H are greatly increased.Under low discharge conditions, the η is higher and the L H is smaller, but the P is larger.Overall, with η as the optimization goal, the η of hydraulic turbines is greatly improved; taking the H as the optimization goal, the TMHT H is greatly increased.That is, it is reasonable to optimize the TMHT performance based on intelligent optimization algorithms and BPNN.

| CONCLUSION
To achieve rapid optimization of TMHT runner parameters, a design method for TMHT runner parameters, which is based on ISMA and BPNN, is proposed.Firstly, the η characteristic neural network (ECNN) and H characteristic neural network (HCNN) are constructed by combining the orthogonal test-based sample data with BPNN, which take the geometric parameters of the runner as input, and the η and H as outputs, respectively.Then, the SMA is improved, and the optimal runner geometric parameters are obtained combining ECNN, HCNN, and improved SMA, achieving rapid optimization of TMHT performance.The conclusions obtained are as follows: samples, this method provides important guidance for optimizing hydraulic turbine parameters.

APPENDIX A: TRAINING DATA OF BPNN AND VERIFICATION OF NUMERICAL SIMULATION
The training data of ECNN and HCNN are shown in Table A1.
To verify the reliability of the numerical simulation, external characteristic tests are conducted on the studied TMHT model.The test bench is shown in Figure A1.The high-pressure inlet pressure of TMHT is provided by three booster pumps, which can be connected in series or parallel to meet different inlet flow conditions.The dynamometer is used to collect the torque and speed of the runner.Pressure sensors are installed at the inlet and outlet to collect pressure data.When conducting external characteristic tests on TMHT, the H, shaft power P, and η under different operating conditions can be obtained by measuring all data.The calculation formulas are shown in Equations (A1)-(A3).

Order Input variables
Output variables where P in is the total inlet pressure of volute; P out is the total pressure at the outlet of the draft tube; ρ is the density of water to be recovered; and M is shaft torque.The TMHT performance is simulated and calculated using Fluent software, and its performance parameters are compared with test data.The results are shown in Figure A2.From Figure A2, it can be seen that the performance trends obtained through experiments and numerical simulations are consistent.Due to the lack of consideration for mechanical loss and leakage in numerical simulation, the η is higher than that of experimental data.At the highest η point, the H error is 2.35%, the η error is 3.08%, and the P error is 6.8%.Overall, the numerical calculation method adopted in this article is suitable for predicting TMHT performance.

F I G U R E 1
Water flow velocity triangle at inlet and outlet of the runner.F I G U R E 2 Structure of ECNN and HCNN.Where F norm (x) is the normalization function, shown as Equation (8); w 11 …, w 5n , w 1 …, w n represent weight value; b 1 …, b i …, b n + 1 represent bias value; m 1 …, m i …, m n represent variables of hidden layer output calculation function, m n + 1 represents variables of output layer output calculation function; f 1 and f 2 are the activation function of hidden layer and output layer respectively, shown as Equations (

F I G U R E 3
Modeling effect of TMHT based on BPNN.T A B L E 1 RMSE and R of ECNN and HCNN.

2 [ 4
Three-dimensional surface diagrams of the four benchmark functions.geometric parameters of the runner are D 1 = 0.51 m, B 1 = 0.045 m, β 1 = 140°, D 2 = 0.26 m, Z = 15, and the water head can reach 542.4 m.The fitness function change curves of ISMA-based ECNN (ISMA-ECNN) and ISMA-based HCNN (ISMA-HCNN) during the training process are shown in Figure 7.

F I G U R E 7
Fitness curves of ISMA-ECNN and ISMA-HCNN.F I G U R E 8 Schematic diagram of TMHT calculation domain model.F I G U R E 9 Grid model of TMHT.

1 .
The BPNN-based TMHT η characteristic neural network and H characteristic neural network models have high accuracy, which shows the modeling feasibility based on BPNN. 2. The improved SMA has strong search efficiency and the ability to avoid falling into local optimum compared to other optimization algorithms.3. Based on ISMA-BPNN, optimizing the geometric parameters of the runner greatly improves the TMHT performance.In the case of sufficient training F I G U R E 10 Verification of independence of grid.F I G U R E 11 Relationship between H and η.F I G U R E 12 Relationship between L H and P.

5
L E A1 ECNN and HCNN training data.