Optimization design for tandem cascades of compressors based on adaptive particle swarm optimization

ABSTRACT To improve the flow performance of tandem cascades on design and off design incidence angle and increase the stable operation range, an optimization system for tandem cascades was developed based on an adaptive particle swarm optimization (APSO-PDC). Firstly, APSO-PDC was proposed based on adaptive selection of particle roles and population diversity control. The adaptive selection of particle roles which combines the evolutionary state and dynamic particle state estimation (DPSE) method will sort the particles into three roles to help different particles execute different search tasks during optimization process. The population diversity control, which combines comprehensive learning strategy of the comprehensive learning particle swarm optimizer (CLPSO) with evolutionary state, pretty strengthens the exploration ability and avoids falling into the local optima. The performance of APSO-PDC is evaluated by 11 unimodal and multimodal functions. Compared with the other six PSOs, the results indicate APSO-PDC has better performance in terms of algorithm accuracy and algorithm reliability. In addition, APSO-PDC is validated by optimizing two large-turning tandem cascades, including low-dimension (5 optimization variables) and high-dimension problems (34 optimization variables). Compared with the other six PSOs, the optimization results demonstrate APSO-PDC has the fastest convergence speed and simultaneously controls well the population diversity.


Introduction
Aerodynamic design of modern compressors faces more and more challenges, with increasing performance requirements for compressors to realize higher efficiency, higher pressure ratio and a more extensive stability margin. As an effective high-load cascade of compressors, tandem cascades have excellent performance in terms of higher stage pressure ratio, fewer stages number and higher stages efficiency compared with similar single cascades (Bammert & Beelte, 1980;Hasegawa, Matsuoka, & Suga, 2003). In addition, tandem cascades have successfully been applied in the compressor stators of engines, such as J85, JT8D and GE J-79 (Lakshminarayana, 1985).
Because of the high load and low loss of tandem cascades, there has been much research on tandem cascades based on the wind tunnel experiment and the computational fluid dynamics (CFD) method. Based on the results of low-speed wind tunnel experiments, Schneider and Kožulović (2013) and Hoeger, Baier, Fischer, and Neudorfer (2011) demonstrated that, compared with single cascades with the same parameters, tandem cascades can provide larger load and smaller loss. Bammert and CONTACT Zhaoyun Song szylast@126.com Staude (1980) investigated the relationship of the front and aft airfoil of tandem cascades, and demonstrated the axial overlap (AO) of 0 or a small negative value can achieve optimal performance of tandem cascades. Roy (1996, 1997) studied the tandem cascades composed of controlled diffusion airfoils (CDA) and the same parameter CDA cascades. The research showed that on design incidence angle, the tandem cascades had larger turning flow angle and lower total pressure loss, but the tandem cascades had poor performances off design incidence angle. Based on low-speed wind tunnel experiments, Heinrich, Tiedemann, and Peitsch (2017) researched the effects of AO and percent pitch (PP) on the performance of tandem cascades, and found tandem cascades obtained the best performance at a high PP and equal load between front and aft airfoil. Based on cascade wind tunnel testing and the CFD method, Hertel, Bode, Kožulovic, and Schneider (2013) studied subsonic and transonic tandem cascades, and found at cascade wind tunnel conditions (at the inlet Mach number of 0.175), the subsonic and transonic tandem cascade work better than the corresponding single cascade with the similar geometry. Dehkharqani, Boroomand, and Eshraghi (2014) applied the CFD method to research a subsonic tandem cascade and the corresponding single cascade, and found the tandem cascade provided higher load and smaller loss. Sachmann and Fottner (1993) found a higher PP can improve the performance of tandem cascades on design incidence angle, but will deteriorate the performance of off-design incidence angle, and a PP of 0.5 can achieve a good balance of the tandem cascades performance on design and off design incidence angle. Based on the CFD method, McGlumphy, Ng, Wellborn, and Kempf (2007) investigated the effect of AO and PP on the performance of subsonic tandem cascades.
They indicated a small AO and a large PP can obviously decrease the flow loss of subsonic tandem cascades on design incidence angle. Compared with similar single cascades with the same load, subsonic tandem cascades can obtain a better flow performance on design and off design incidence angle. Based on the literature, tandem cascades can realize a higher stage pressure ratio and stages efficiency, which is significant in the design of advanced compressors. However, some literature has indicated the stability margin of compressors with tandem cascades is smaller than that of compressors with similar single cascades. Based on the experimental method, Eshraghi, Boroomand, and Tousi (2014) found that the isentropic efficiency and stability margin of the high-load tandem rotor are lower than those of single row rotor with the same parameters. Brent, Cheatham, and Clemmons (1972) studied a tandem rotor of core compressors by experimental testing, and found that the pressure ratio of the tandem rotor exceeded the design pressure ratio, but the tandem rotor had smaller isentropic efficiency and stability margin. Tandem rotors were used on multistage axial compressors by Staude (1980, 1981) and Bammert and Beelte (1980). On design point, the isentropic efficiency of the tandem rotor compressor was 0.856, and the stability margin was only about 5%.
It is well known that CFD has greatly improved the optimization design method, combining CFD and optimization algorithm. Hence, much research has been carried out on the optimization design method based on CFD. Ezhilsabareesh, Rhee, and Samad (2017) accomplished the shape optimization of a turbine based on multiple surrogates-assisted and multi-objective evolutionary algorithms. Optimization of vanes of centrifugal compressors is explored in Ivo, Damir, and Zoran (2016) by using control points to parameterize the suction and pressure surface of vanes. An optimization design method for radial turbines was proposed by Galindo, Hoyas, Fajardo and Navarro (2014), and a CFD-based hull form optimization loop was developed by combining an approximate method and an improved particle swarm optimization (PSO) algorithm in Zhang, Zhang, Tahsin, Leping, and Yu (2017). A multi-objective optimization method for wing is presented in Liang, Cheng, Li, and Xiang (2014). Safikhani, Khalkhali, and Farajpoor (2014) applied CFD and meta-models to accomplish the multi-objective optimization design of centrifugal pumps. Thus, to improve the flow performance of tandem cascades on design and off design incidence angles, an optimization system for tandem cascades was developed in this study.
In the literature about optimization of tandem cascades, the relative positions of the front airfoil and rear airfoil have been used as optimization variables. However, the shape of tandem cascades has a great influence on the flow field. Therefore, the present study focuses on the coupling optimization for shape and relative position of tandem cascades. In contrast to previous studies, with simultaneous variations of shape and relative position the number of optimization variables and the complexity of optimization will greatly increase. Hence, the major difficulty is the development of an optimization method that has fast convergence speed and simultaneously obtains an optimal solution. Thus, based on the above analysis, the main contribute of this study is twofold. The first goal is to propose a new variant of PSO, namely adaptive particle swarm optimization with population diversity control (APSO-PDC), to improve the performance of the original PSO. The second goal is to create an automatic optimization system of tandem cascades based on APSO-PDC to improve the design quality of tandem cascades. The optimization system can be used to realize the optimization design of the shape parameters and relative position of tandem cascades.

Related work of PSO
Over the past few decades, many real-life applications in different fields have been successfully solved based on contemporary soft computing techniques. Sefeedpari, Rafiee, Akram, Chau, and Pishgar-Komleh (2016) proposed an adaptive neural fuzzy inference systembased modeling approach where the number of data pairs employed for training was adjusted by application of a clustering method. Wang, Xu, Chau, and Chen (2013) combined a data analysis methodology and support vector machine to decompose annual rainfall series. Gholami, Chau, Fadaee, Torkaman, and Ghaffari (2015) studied the groundwater level fluctuations based on an artificial neural network. Chen and Chau (2016) presented a hybrid double feed forward neural network model. Among many contemporary soft computing techniques, evolutionary algorithms have become an important research field. PSO is one of the most popular and effective swarm intelligence algorithms, and was developed by Kennedy and Eberhart based on the natural phenomenon of birds looking for food (Kennedy & Eberhart, 1995). As PSO is easy to implement, the parameter needed to define is small and the convergence rate is fast, it has been studied by many scholars since it was first proposed. Many studies have shown that some PSOs easily find local optima when optimizing complex problems. Ratnaweera, Halgamuge, and Watson (2004) proposed that the main reason for the premature convergence of PSOs is that the population diversity is too simple. However, increasing the population diversity may lead to a slower convergence speed. Thus, the two most important goals of PSO research are to speed up convergence and avoid falling into the local optima. Some PSO variants have been proposed to achieve these two goals. However, according to the literature, it is difficult to achieve these two objectives simultaneously.
In original PSO, each particle has two parameters, the current position x i and the current velocity v i . During iteration, PSO will memorize the individual optimal position (pbest) and the global optimal position (gbest). At each iteration, the following formulas are used to recalculate the velocity and position of each particle.
where w is the inertial factor, C 1 and C 2 are the study factor, r 1 and r 2 are the random numbers uniformly generated from [0, 1], N is the size of population, D is the number of optimization variables. A limit velocity called v max is imposed on particles. According to the literature on PSO variants, in order to accelerate convergence speed and avoid falling into the local optima, PSO was improved with the following four methods.
(1) Control of population diversity: negative entropy is introduced to PSO in Xie, Zhang, & Yang (2002) to avoid premature convergence. To increase population diversity, a method of adaptively choosing the optimal position of neighborhood was raised in Li (2004). A hierarchical clustering method was used to locate and track multiple peaks for dynamic optimization problems at each iteration (Yang & Li, 2010). Based on these studies, these PSOs improve the diversity of the algorithm and also reduce the convergence speed of PSO.
(2) Change of population topology: the population topology of PSO determines the method to share information among population. A PSO variant called FIPS (The Fully Informed Particle Swarm Optimization) in (Mendes, Kennedy, & Neves, 2004) making the individuals "fully informed" was proposed. FIPS used an average of the individual optimal position of its neighbors as the individual optimal position (pbest). Liang, Qin, Suganthan, and Baskar (2006) presented the comprehensive learning particle swarm optimizer (CLPSO); CLPSO uses a novel learning strategy in which all individual optimal positions are used to update the particle's velocity. Wang, Yang, and Chen (2014) proposed multi-layer particle swarm optimization (MLPSO) to improve the performance of traditional PSO, which consisted of only two searching layers. PSO-ITC (Particle swarm optimization with increasing topology connectivity) (Lim & Mat Isa, 2014) applied an ITC module to increase population connectivity. In summary, these methods can remarkably enhance the performance of PSOs on some problems. (3) Hybrid PSO: Hybrid evolutionary algorithms are becoming increasingly popular due to their capabilities in dealing with complexity problems. Zhang and Xie (2003) proposed a PSODE (hybrid particle swarm with differential evolution operator) algorithm, which integrates particle swarm optimization with differential evolution. A cooperative PSO (CPSO) algorithm was proposed in Vanden-Bergh and Engelbrecht (2004) (Liu, Qin, & Li, 2007) was proposed based on species-PSO and an adaptive species radius strategy for multimodal function optimization. PSO with adaptive parameters, called adaptive PSO (APSO), has been proposed by Zhan, Zhang, Li, and Chung (2009). In APSO, four evolutionary states, exploitation, exploration, convergence, and jumping out are defined. Similarly, the four operators in ALPSO (Self-Learning Particle Swarm Optimizer) (Li, Yang, & Nguyen, 2012) play similar roles as the four evolutionary states defined in APSO.

APSO-PDC
In the literature about PSOs, all particles in PSOs use the same parameters, inertia factor and study factor, which may cause a lack of particle diversity. Hence, according to their local fitness and local locations, different particles should use different parameters to execute different searches including local search and global search. Inspired by the idea of division of labor, different roles can be assigned to a particle, for example, convergence particle, exploiting particle and exploring particle, which will execute different search tasks during the optimization process. In order to maintain population diversity and simultaneously preserve good particle information, we use a strategy based on the comprehensive learning strategy of CLPSO and evolutionary state to control population diversity. Based on the aforementioned analysis, to effectively raise the performance of PSO, in this paper, APSO-PDC is formulated based on adaptive selection of particle role and population diversity control. In this paper the following discussion is restricted to minimization problems.

Adaptive selection of particle roles
Inspired by the division of labor, we can assign different roles to a particle, for example, convergence particle, exploiting particle and exploring particle, which will execute different search tasks during the optimization process. To adaptively select an optimal role from these three roles for each particle, an adaptive selection role method is created based on evolutionary states (ES) and the dynamic particle state estimation (DPSE) function. According to APSO (Zhan et al., 2009), four evolutionary states, including exploitation, exploration, convergence, and jumping out, are defined based on evolutionary factor fe (defined in Equation (2)). In APSO, fuzzy classification was adopted to classify evolutionary factor into four evolutionary states, and the fuzzy membership functions are depicted in Figure 1. On the other hand, to obtain the quality of particles and decide the optimal role for a particle from the three roles, a DPSE method is first devised based on local fitness and local location of particles. The DPSE function is defined by Equation (3). where In Equation (2), g i is the mean distance of particle i to all the other particles. g g is the mean distance of the global best particle. And n is the size of population, D is the number of optimization variables. fe is the evolutionary factor. In Equation (3), r i and f i represent the local location and local fitness of particles, respectively. DPSE function is the dynamic particle state estimation function.
As can be seen from Equation (3), particles with small local fitness may be the local or global optima particle, and particles with small local position are very near to the best position. Therefore, particles with small DPSE should perform a local search around the local or global optimal region, and particles with large particle fitness and large position will perform a global search to explore as many optima as possible during the optimization process.
According to DPSE, we can select an optimal role for a particle from the three roles. Convergence particles with small DPSE, which are around the best particle, will be used to perform further searches around the globally optimal region. The role of exploiting particles whose positions and fitness are both around the better particle or the local optima will help achieve further searches around the local optimal region. Exploring particles with large DPSE, whose positions are far away from the best particle or whose fitness is far worse than that of the best particle, will be used to explore as many optima as possible during the optimization process.

Adaptive control of parameters
In PSO, the value of individual study factor (c1) makes individual particles go around their historical best position and helps explore local optima. The value of global study factor (c2) helps the swarm converge to the current global best region. Therefore, particles with different roles should have different values of c1 and c2 to execute different searches including local search and global search. For the convergence particle, C1 = C1min, C2 = C2max. Convergence particles are used to perform further searches around the globally optimal region; thus, the value of C2 should be equal to the maximum. Exploiting particles are used to further search around the local optimal region. Hence, a proper C1 and C2 can simultaneously emphasize the local search and convergence speed. Exploring particles are used to explore as many optima as possible during the optimization process. Hence, a large C1 and a small C2 can help exploring particles explore more individual best positions. Hence, for adaptive control of parameters, we dynamically change C1, C2 and W. We create Algorithm 1 based on evolutionary states and the aforementioned analysis. When the evolutionary state is exploitation state or exploration state, bigger C1min and C1max can help exploiting particles make full use of the local optimal information and exploring particles find more optima. For particles of each role, the value of C1, C2 and W are calculated in Algorithm 1. In Algorithm 1, after many experiments, the initial values of c1min, c1max, c2min, c2max, wmin, and wmax are 1.0, 1.8, 1.0, 1.8, 0.3, and 0.7, respectively. Because the above adjustments on the parameter values should not be too irruptive, the maximum increment or decrement is Rd, which is called the "acceleration rate" in this paper. The default value of Rd is 0.05.

Population diversity control
Many studies have shown that some PSOs easily find local optima when optimizing complex problems. Ratnaweera et al. (2004) proposed that the main reason for the premature convergence of PSOs is that the population diversity is too simple. Hence, it is important to research population diversity control. To control population diversity, we introduce a new strategy based on the comprehensive learning strategy of CLPSO (Liang et al., 2006) and evolutionary states to update pbest. In PSO, the fitness of the particle is determined by the values of all independent variables, and a particle with some independent variable that gets a better value may have a low fitness due to the influence of other bad independent variables. In order to make better use of good independent variables, a new learning strategy was proposed in CLPSO. However, CLPSO focuses on increasing the population diversity, but results in a slower convergence. Hence, to accelerate convergence speed and keep population diversity, we introduce a control parameter Mp, which is mutation probability related to the ES. In summary, we develop a procedure to update pbest based on CLPSO and ES. The procedure is described in Algorithm 2. For each particle, we use Algorithm 2 to update the individual optimal position of particles. When ES is convergence and jumping out, the algorithm executes a global search and it is easier to obtain local optima. Hence, a small Mp not only keeps a fast convergence speed but also avoids premature convergence. On the other hand, when the ES is exploitation and exploration, the algorithm achieves further search around the local optimal region and explores as many optima as possible; Mp equals Mp max to increase the population diversity. Mpmax and Mpmin are the maximum and minimum mutation probability. The values of Mpmax and Mpmin are 0.4 and 0.1, respectively.

Flowchart of APSO-PDC
Based on the aforementioned analysis, the flowchart of APSO-RPDC is summarized in Algorithm 3.

Test functions and parameter settings
In order to verify the effectiveness of APSO-PDC, further experimental tests with benchmark functions were carried out to assess the performance of APSO-PDC and to compare APSO-PDC with some PSO variants. These test functions, which are shown in Table 1, including unimodal functions: F1-F6, multimodal functions: F7-F11.
The parameter settings for the PSO variants used are given in Table 2. To fairly compare the optimization performances of APSO-PDC and the other six PSOs and eliminate the error caused by randomness, seven PSOs were run independently 30 times on the 11 test problems. During the test process, the dimension of functions was 30, and the number of population was 40.

Comparisons of solution accuracy
In order to assess the solution accuracy of the seven PSOs, in this test, the stop criterion of the seven PSOs was that the algorithm reached the maximum number of the function evaluations, that is 2e5. The performance of the seven PSOs was assessed using the mean fitness value (Fmean) and the standard deviation (Std). Table 3 shows the mean fitness value (Fmean) and standard deviation (Std) of the optimal value of the 11 test functions obtained by 30 independent runs. The values of Fmean and Std were average values of 30 independent runs. The best results from the seven PSOs are indicated by boldface in the table. Table 3 shows that APSO-PDC achieves the best performance on the majority of the problems, except function f4. Specifically, only APSO-PDC can successfully find a better optimum for f8 and f9. Hence, APSO-PDC exhibits the outstanding exploration ability. This attribute is mainly due to the population diversity control of APSO-PDC that enhances the global searching ability of APSO-PDC. Figure 2 gives the convergence process of the seven different PSOs on some test functions. It can be seen that APSO-PDC apparently has a faster convergence speed than the other algorithms. Figure 2 shows that APSO-PDC not only increases the population diversity but also results in a faster convergence.

Comparisons on convergence speed and reliability
To compare the convergence speed and reliability of the seven PSOs, in this test, the stop criterion of the seven PSOs was to obtain acceptable values of the test functions, which are presented in Table 1. If the number of function

Algorithm
Year Topology PSO-W (Shi & Eberhart, 1998) 1998 Global star FIPS (Mendes et al., 2004) 2004 Local Uring CLPSO (Liang et al., 2006) 2006 Comprehensive Learning APSO (Zhan et al., 2009) 2009 Global star FLPSO-QIW (Tang, Wang, & Fang, 2011) 2011 Comprehensive learning MLPSO-STP (Yu, Wang, & Wang, 2016) 2016 Fully connected     Table 4 shows the mean value of the SR and the MFE of 30 independent runs. 'NA' means that no runs reached the acceptable value in Table 4. SR is an important performance parameter showing algorithm reliability, which is the ratio of the number of successful runs over the total number of runs. A successful run means that the algorithm obtains the acceptance value within the maximum number of fitness evaluations. Boldface in Table 4 indicates the best result among those obtained by all algorithms. As can be seen from Table 4, for all algorithms, APSO-PDC has the fastest convergence speed on all functions, and APSO ranks second. Moreover, APSO-PDC shows an excellent search reliability among all used PSOs, according to comparing the SR in Table 4. Table 4 also indicates that APSO-PDC has the highest SR at 99.52%, and MLPSO-STP ranks second. This excellent performance on all functions proves that APSO-PDC can accelerate convergence speed and simultaneously avoid falling into the local optima.

Parameter sensitivity analysis of APSO-PDC
There are nine important parameters in APSO-PDC: c1min, c1max, c2min, c2max, wmin, wmax, Mpmin, Mpmax and Rd. The default values of these parameters are 1.0, 1.8, 1.0, 1.8, 0.3, 0.7, 0.1, 0.2 and 0.05, respectively. To find out how these important parameters affect the performance of APSO-PDC, an experiment on the parameter sensitivity analysis of APSO-PDC was conducted on the 11 test function in 30 dimensions. To separately test the effect of a particular parameter, we only changed one parameter in one experiment and used the default values of the other parameters. Table 5 shows the mean fitness value (Fmean) of the 30 independent runs of three functions: unimodal functions f3, multimodal functions f7and f9. The best results of Fmean are indicated by boldface in the table. As can be seen in Table  5, for c1min, c1max, c2min, c2max, wmin and wmax, these parameter values are so reasonable that rational destabilization of these parameters only changes a little for the optima obtained by APSO-PDC, and these parameters may use the default value in real application. Hence, there are only three important parameters in APSO-PDC: the maximum and minimum of mutation probability (Mpmin, Mpmax), and the acceleration rate (Rd). For multimodal functions and unimodal functions,

Tandem cascades
To further compare the performance of the PSO variants and improve the design quality of tandem cascades, we applied the PSOs to create optimization systems to obtain the best tandem cascades. The geometric model of tandem cascades is shown in Figure 3. Table 5 displays the  definition of relative parameter of the front and aft airfoil of tandem cascades. As mentioned in the introduction, the five relative parameters about the arrangement of the front and aft airfoil, camber ratio (TR), chord ratio (CR), AO, PP and incidence angle of the rear airfoil (KBB) are important for the design of tandem cascades and are presented in Table 6.

CFD simulations
In this work, NUMECA/AUTOGRID5 was used to generate the structured grid of tandem blade. The steady (CFD) simulations are accomplished by NUMECA FINE/TURBO, which solves the conservative Reynolds averaged Navier-Stokes equations with Spalart-Allmaras turbulence model. In the CFD simulations of CDA and tandem cascades, the constant total temperature (288.15 K), and total temperature (101325 Pa) are uniformly imposed at the inlet. The flow angle and outlet mass flow were changed to obtain the desired incidence angle and inlet Mach number.
In order to obtain the effect of grid cells of tandem cascades on the numerical simulation results, we investigated the effect of the number of grids on the CFD simulations results of the initial tandem cascade on the  design point, namely at the inlet Mach number of 0.7 and the incidence angle of −2°. Table 7 presents the numerical results of the tandem cascade with different grids. The total pressure loss coefficient (Loss) and the static pressure ratio (Pt) of cascades are defined as follows.
In the above formula, P 1 and P 2 are the mass averaged static pressure at the inlet and outlet of tandem cascades, respectively. Similarly, p * 2 and p * 1 are the mass averaged total pressure at the inlet and outlet, respectively.
It can be seen that the Loss and Pt are almost unchanged between Grid 5 and Grid 6. Based on the accuracy and the time taken of CFD simulations, we chose Grid 6 (the grids of span-wise × circumferential × axial: 2 × 197 × 189) as the grid of CFD. The grids of CFD simulations for tandem blades are shown in Figure 4.

Optimization system
To further compare the performance of the PSO variants and improve the design quality of tandem cascades, we applied the PSOs to create optimization systems to accomplish the optimization design of tandem cascades. The flowchart of optimization system is shown in Figure 5. In the first step, the tandem cascades were parameterized. After parameterization, the tandem cascade was expressed by some parameterization variables, and the optimization variables can be obtained from the parameterization variables. Next, we used the PSO variants, indicated by IPSO, to generate initial individuals according to optimization variables. In the third step, the fitness is evaluated, which includes tandem cascades generation, grid generation and flow field calculation based on CFD simulations. In the final step, if the convergence condition is satisfied, the optimization system is stopped.
Otherwise the system will apply the IPSO to search the optimal tandem cascade.

Parameterization method and optimization variables
The goal of this optimization design was to investigate the influence of five configuration parameters, TR, CR, AO, PP and KBB, on the aerodynamic performance of tandem cascades on design and off design incidence angle. Thus, the parameterization of tandem cascades is accomplished by a program of tandem cascades generation. The optimization variables are the five configuration parameters of tandem cascades. Based on the related literature in the introduction and design experiences of tandem cascades, the upper and lower bound about optimization variables was selected; Table 8 shows the upper bound and lower bound of optimization variables.

Initial tandem cascades
An excellent CDA cascade, which is studied in Gelder, Schmidt, Suder, and Hathaway (1987), was selected as the original cascade. An initial tandem cascade was designed to satisfy the main design parameters of CDA, which is described in Gelder et al. (1987). The main design parameters of CDA are presented in Table 9. In order to design an initial tandem cascade with fine flow performance, based on the related literature in the introduction, the value of five configuration parameters of the initial tandem cascades are selected and presented in Table 10.

Optimization objectives and constraints
To improve the flow performance of the tandem cascade at large positive incidence angle, the optimization objective is to reduce the total pressure loss coefficient at the large positive incidence angle, the incidence angle of 3 degrees and the inlet Mach number of 0.7. The constraint is that the static pressure ratio of tandem cascades at the optimization point is not smaller than that of the initial tandem cascades.

Optimization results and discussion
To compare the improved PSO mentioned above, we respectively applied the seven improved PSOs to optimize the tandem cascade. In this optimization, the total number of optimization variables is five. In the optimization, for all algorithms, the number of population is 20, and the maximum iteration number is 200. The stop criterion of all algorithms is that the maximum iteration number is reached. Figure 6 shows the convergence process of all algorithms. Table 10 reveals the optima of the objective function obtained by all algorithms. hen the total number of optimization variables is 5, Table 11 illustrates that APSO-PDC acquires the optimal value of objective function for all PSO variants, and FLPSO-QIW (Feedback learning particle swarm optimization) ranks two. Figure 6 shows that APSO-PDC has the fastest convergence speed and simultaneously controls the population diversity well. During the optimization, at the initial stage, APSO-PDC has good population diversity, and at the last stage, APSO-PDC has an excellent convergence speed. This excellent performance on real problems shows that APSO-PDC holds an appropriate balance between exploitation ability and exploration ability. Tables 12 and 13 respectively reveal the comparisons of aerodynamic performance and the five configuration parameters of the original tandem cascade and the optimal tandem cascade. To describe conveniently, in this paper, ORG-TAN and OPT-TAN respectively represent the original tandem cascade and the optimal tandem cascade obtained by APSO-PDC. After optimization, at the optimization point, namely at the large positive incidence angle, the total pressure loss coefficient of OPT-TAN is decreased by 55%.  Figure 7 shows the comparisons of total pressure loss coefficient of ORG-TAN and OPT-TAN. The total pressure loss of ORG-TAN and OPT-TAN at the incidence angle of 0°is almost unchanged after optimization. OPT-TAN significantly reduces the total pressure loss coefficient at positive incidence angle, but OPT-TAN also worsens the flow performance at negative incidence angle. The following section further discusses the total pressure loss and flow characteristics of tandem cascades at large negative and positive incidence angle. Figures 8 and 9, respectively, show the comparisons of Mach number contours and entropy contours of ORG-TAN and OPT-TAN at the incidence angle of 3°, namely the optimization incidence angle. According to Figures 8  and 9, for the original tandem cascade (ORG-TAN), there is a large-scale separation on the trailing edge of front airfoil. The low-energy fluid and the wake of the trailing edge of the front airfoil are mixed with each other, resulting in a wide range of low-velocity zone and severe mixing loss. After optimization, OPT-TAN significantly reduced the separation losses and the mixing losses of the     Table 12, the main reason that OPT-TAN has a good flow performance is because OPT-TAN reduced the load of the front airfoil by increasing TR. For tandem cascades, the decrease of the gap area achieved by a raise of PP and decrease of AO can obviously improve the flow loss of tandem cascades at positive incidence angle. Figures 10 and 11, respectively, show the comparisons of Mach number contours and entropy contours of ORG-TAN and OPT-TAN at the incidence angle of −6°. Figure 7 illustrates that OPT-TAN worsens the flow performance at negative incidence angle. According to Figures 10 and 11, ORG-TAN has a small low-velocity zone and separation losses on the suction surface of the front airfoil, but OPT-TAN brings a great deal of lowenergy fluid on the cascade passage after the fluid leaves the trailing edge of the front airfoil. Thus, OPT-TAN generates a great many high-entropy zones and low losses on the cascade passage. Based on the above analysis, the increase of TR may increase the total pressure loss of the front airfoil of tandem cascades at large negative incidence angle.

Parameterization method and optimization variables
The goal of this optimization design was to simultaneously investigate the influence of the shape parameters of tandem cascades and two configuration parameters, AO and PP, on the aerodynamic performance of tandem cascades on design and off design incidence angle. As seen in Figure 12, the parameterization of the camber line of the cascades and the parameterization of the thickness distribution of the cascades are accomplished by the NURBS (Non-Uniform Rational B-Splines) method with 10 control points. Compared with the traditional approach based on the polynomial method, the NURBS method can parameterize complex curves with fewer control points. In addition, the NURBS parameterization method can realize partial modification of complex curves and effectively reduce the number of optimization variables. In Figures 12 and 13, CH represents the chord of the cascades, and TM represents the relative thickness of the cascades. The parameterization of two configuration parameters, AO and PP, is accomplished by a program of tandem cascades generation.   During the optimization, the first and the last of the control points of the NURBS curves remain unchanged, and the other eight control points of NURBS of the camber line and the thickness distribution and the two configuration parameters AO and PP were selected as optimization variables of the tandem cascade. AO and PP vary within the range of [−0.1, 0.1] and [0.6, 0.95], respectively. In the process of NURBS parameterization, the irregular motion of control points may result in unreasonable tandem cascades. Therefore, in this study, the motion direction of control points is defined along the vertical direction of the NURBS curves, so that each of the control points can be described by a vertical coordinate. In the optimization, the total number of optimization variables is 34.

Initial tandem cascades
In this work, the optimization system was used to optimize a large-turning tandem cascade, which was obtained from the hub of an excellent tandem stator. The basic parameters of the initial tandem cascade are shown in Table 14. In Table 14, C, S and σ are applied to express the chord length, the pitch width and the solidity of tandem cascades, respectively. The optimization point of this tandem cascade is the design point, the incidence angle of 2 degree and the inlet Mach number of 0.8.

Optimization objectives and constraints
To improve the flow performance of the tandem cascade on design point, the optimization objective is to reduce the total pressure loss coefficient at the incidence angle of 2 degree and the inlet Mach number of 0.8. The constraint is that the static pressure ratio of tandem cascades at optimization point is not smaller than that of initial tandem cascades.

Optimization results and discussion
To compare the improved PSO mentioned above, we also respectively apply the seven improved PSOs to optimize the tandem cascade. In the optimization, for all algorithms, the number of population is 40, and the maximum iteration number is 200. The stop criterion of all algorithms is that the maximum iteration number is   reached. Table 15 reveals the optimal values of the objective function obtained by all algorithms. Figure 14 shows the convergence process of all algorithms. When the total number of optimization variables is 34, Table 15 shows that APSO-PDC acquires the optimal value of objective function for all PSO variants, and MLPSO-STP ranks two. In addition, the objective function obtained by APSO-PDC is decreased by 14.7% compared with MLPSO-STP. Figure 14 shows that APSO-PDC has the fastest convergence speed and simultaneously controls the population diversity well. This excellent performance on real problems shows that APSO-PDC holds an appropriate balance between exploitation ability and exploration ability. Table 16 reveals the comparisons of aerodynamic performance of the original tandem cascade and the optimal tandem cascade. For conveniently description, in this paper, ORG-TAN and OPT-TAN, respectively, represent the original tandem cascade and the optimal tandem cascade obtained by APSO-PDC. After optimization, compared with ORG-TAN, at the optimization point, namely at the incidence angle of 2 degrees, the total pressure loss coefficient of OPT-TAN is decreased by 51%. Figure 15 shows the changes of total pressure loss coefficient along with incidence angles of the original   (ORG-TAN) and the optimal (OPT-TAN) tandem cascade, obtained by APSO-PDC. Figure 15 shows that at the inlet Mach number of 0.8, the total pressure loss coefficient of OPT-TAN was smaller than that of ORG-TAN at all incidence angles. Figures 16 and 17, respectively, show the comparisons of the geometry and the thickness distribution of ORG-TAN and OPT-TAN. After optimization, the gap area of OPT-TAN is reduced by an increase of PP. For the front airfoil, the relative thickness was increased in the 5% to 50% relative position of axial chord length, and was decreased in the 50% to 80% relative position of axial chord length. For the aft airfoil, the relative thickness was decreased in the 20% to 80% relative position of axial  chord length. Further analysis of flow performance of tandem cascades is shown in the following paragraphs. Figures 18 and 19, respectively, show the comparison of the Mach number contours and entropy contours of ORG-TAN and OPT-TAN at the incidence angle of 2 degrees. Figures 18 and 19 show that OPT-TAN has a smaller wake low-velocity region of the front airfoil, which obviously reduces the mixture loss of the tandem cascade. In addition, OPT-TAN obviously reduces the boundary layer thickness of the suction surface of the cascade, which reduces the friction loss of the boundary layer. The gap area of OPT-TAN was smaller than that of ORG-TAN, which decreased the flow fluid through the gap. The mixing loss generated by the mixture of the gap fluid with the mainstream fluid may be decreased because of the decrease of the gap fluid, which is also illustrated in Figure 19. In Figure 19, the high-entropy zone of the passage of OPT-TAN is smaller that of ORG-TAN, which demonstrates the total pressure loss of TAN-TAN is also smaller that of ORG-TAN. Figures 20 and 21, respectively, show the comparison of the Mach number contours and entropy contours of ORG-TAN and OPT-TAN at the incidence angle of −2 degrees. In Figure 20, the high Mach number region of the front airfoil of OPT-TAN is obviously reduced   after optimization, which decreases the shock loss. In addition, the low-velocity region on the suction surface of OPT-TAN was also decreased. Hence, the loss of OPT-TAN was obviously reduced, which is demonstrated in Figure 21. In Figure 21, in the whole cascade passage of OPT-TAN, the high-entropy zone of ORG-TAN is obviously bigger than that of OPT-TAN. Figure 22 shows the comparison of the entropy contours of ORG-TAN and OPT-TAN at the incidence angle of 4 degrees. Figure 22 reveals the high-entropy zone of OPT-TAN is obviously smaller than that of ORG-TAN, which demonstrates that the flow performance of TAN-TAN is better that of ORG-TAN.

Conclusion
To improve the flow performance of tandem cascades on design and off design incidence angle and increase the stable operation range, an optimization system for tandem cascades was developed based on an APSO-PDC.
Firstly, APSO-PDC was proposed based on adaptive selection of particle roles and population diversity control. The adaptive selection of particle roles, which combines the ES and DPSE method, will sort the particles into three roles to let different particles execute different search tasks during the optimization process. The population diversity control, which combines the comprehensive learning strategy of CLPSO with ES to update the individual optimal position, strengthens the exploration ability and avoids falling into the local optima of APSO-PDC. The performance of APSO-PDC is comprehensively evaluated by 11 unimodal and multimodal functions. Compared with the other six PSOs, the results indicate that APSO-PDC has better performance in terms of algorithm accuracy and algorithm reliability.
In addition, APSO-PDC is validated by optimizing two large-turning tandem cascades, including low-dimension (five optimization variables) and highdimension problems (34 optimization variables). Compared with the other six PSOs, the results demonstrate APSO-PDC has the fastest convergence speed and simultaneously controls the population diversity well.
After optimization design of tandem cascades, at the optimization point, the total pressure loss coefficient of the optimal cascade is decreased by 55% for the lowdimension case and 51% for the high-dimension case. An increase of TR can effectively improve the flow performance of tandem cascades at positive incidence angle, and may increase the total pressure loss of the front airfoil of tandem cascades at large negative incidence angle. The decrease of the gap area achieved by an increase of PP can obviously reduce the flow loss of tandem cascades at positive incidence angle.
The current study shows encouraging results and represents a foundation for further study. Planned future work will include multi-objective optimization design and analysis of tandem cascades, 3D optimization design of tandem blades, and experimental validation of tandem cascades and 3D blades.