Abstract
Engine calibration aims at simultaneously adjusting a set of parameters to ensure the performance of an engine under various working conditions using an engine simulator. Due to the large number of engine parameters to be calibrated, the performance measurements to be considered, and the working conditions to be tested, the calibration process is very timeconsuming and relies on the human knowledge. In this paper, we consider nonconvex constrained search space and model a real aeroengine calibration problem as a manyobjective optimisation problem. A fast manyobjective evolutionary optimisation algorithm with shiftbased density estimation, called fSDE, is designed to search for parameters with an acceptable performance accuracy and improve the calibration efficiency. Our approach is compared to several stateoftheart many and multiobjective optimisation algorithms on the wellknown manyobjective optimisation benchmark test suite and a real aeroengine calibration problem, and achieves superior performance. To further validate our approach, the studied aeroengine calibration is also modelled as a singleobjective optimisation problem and optimised by some classic and stateoftheart evolutionary algorithms, compared to which fSDE not only provides more diverse solutions but also finds solutions of highquality faster.
Introduction
Multiobjective optimisation has been widely applied to diverse realworld problems [12, 24], while manyobjective optimisation problems are underexploited but commonly seen in real life [17]. An important example is the engine calibration within car engine design [28].
Engine calibration aims at simultaneously adjusting a set of parameters to ensure the performance of an engine under various working conditions using an engine simulator. A working condition is often described by a number of parameters depending on the actual usage of engine, such as the speed. Traditional calibration process is achieved by manually changing the engine parameters, simulating its state under different working conditions, comparing its states on engine maps [23], and repeating the above steps until the states are closed to some desired states. Figure 1 illustrates the manual process of adjusting engine parameters. The calibration process is usually timeconsuming and highly relies on the human knowledge. Ma et al. [22, 23] proposed a multiobjective optimisation approach to automate the gasoline direct injection engine calibration process, in which five engine parameters were adjusted under four cases and three evaluation criteria were considered for each case. However, some of the engines, such as aeroengines, are controlled by dozens of parameters. To achieve optimal performance, the engines should be examined under more working conditions. Due to the large number of engine parameters to be adjusted, the performance measurements to be considered, as well as the working conditions to be tested, the calibration process is a crucial manyobjective optimisation problem.
In this work, we consider a real aeroengine with nonconvex search space of 27 parameters (e.g., compressor pressure ratio, compressor efficiency, and combustion efficiency) and its expected performance under 33 different flight conditions. An engine state is described by 15 measurements, such as compressor exit pressure and fuel flow parameters. Therefore, 27 parameters need to be adjusted at the same time to make the engine’s states as closed to the given measurement data as possible.
We model this calibration process as a manyobjective optimisation problem. A fast manyobjective evolutionary optimisation algorithm with shiftbased density estimation is designed to search for parameter setting with acceptable performance accuracy and improve the calibration speed. During optimisation, a numerical simulation program is used to simulate the performance of the engine given a parameter setting and a working condition. Our approach is compared to several stateoftheart many and multiobjective optimisation algorithms on this realworld engine calibration problem and achieves superior performance.
The main contributions of this paper are as follows.

Efficient calibration of a real aeroengine: a real aeroengine model and real data of performance measurements are used in this work.

Large parameter search space: We consider 27 parameters of this aeroengine. To our best knowledge, no existing work has ever calibrated a such big number of parameters.

Large number of criteria to be optimised: Four hundred and ninetyfive criteria are considered in our aeroengine calibration process. To our best knowledge, no existing work on engine calibration has ever considered a such large number of criteria to be optimised.

Nonconvex constrained parameter space: The calibration of model parameter is a constrained optimisation problem. The parameter space of our aeroengine model is nonconvex and blackbox. Given a parameter setting, the computational fluid dynamics (CFD) simulation of engine dynamics is able to determine if a given parameter setting is feasible or not, but it is timeconsuming as an evaluation of a parameter setting.


Two optimisation approaches for automating the calibration process:

Manyobjective optimisation with a simple yet effective fSDE: We model the calibration process as a manyobjective optimisation problem and propose a fast manyobjective evolutionary algorithm with Shiftbased Density Estimation (fSDE) for automating and speeding up the calibration process. The proposed fSDE shows its superior or competitive performance to some stateoftheart many and multiobjective evolutionary algorithms on our real aeroengine calibration problem and a wellknown manyobjective optimisation benchmark test suite.

Singleobjective optimisation: To further validate our approach, the process is also modelled as a singleobjective optimisation problem with a linear aggregation of criteria to be optimised. Several classic and stateoftheart evolutionary algorithms have been applied, but their efficiency is worse than our proposed fSDE, probably due to the infeasible parameter settings.


Inspiration to human engineers:

Fault diagnosis assistant: It has been observed that very different engine parameter settings lead to similar performance, which motivates human engineers to seek for the reasons and understand why it happened (e.g., sensibility of parameter, faults, or bias in computational model).

Additional parameters and measurements: As different settings lead to similar performance given the current parameters, measurements, and operation points, human decision makers can consider additional calibration parameters, measurements, or operation points.

Novelty of design: From a perspective of discovering engines, instead of problem solving, diverse settings found by fSDE inspire human engineers to design novel engines.

In the remainder of this paper, “Background” discusses the background and related work. “A realworld aeroengine calibration problem” presents the real aeroengine calibration problem studied in this paper. The many and singleobjective evolutionary optimisation for calibration are provided in “Manyobjective evolutionary optimisation for aeroengine calibration” and “Singleobjective evolutionary optimisation for aeroengine calibration”, respectively. In particular, “fSDE: fast MaOEA with shiftbased density estimation” introduces our proposed fSDE. Our fSDE is compared to some stateoftheart many and multiobjective evolutionary algorithms on a wellknown manyobjective optimisation benchmark test suite (“Efficiency verification of fSDE on benchmark test suite”) and our real aeroengine calibration problem (“fSDE for aeroengine calibration”). Finally, “Conclusion” concludes.
Background
Engine calibration
Engine calibration refers to the process of adjusting a set of parameters of a given engine model, denoted as \(\varvec{x}=(x_1, x_2,\dots , x_d)\), to ensure its performance at several operation points or working conditions (illustrated in Fig. 1). An operation point s is often determined by a number of values that describe the actual usage of the engine. Taking aeroengines as an example, an operating point is specified by a Mach number, a flight altitude, and a flight state (e.g., takingoff and cruising), which is quantified by a mechanical spool speed [14]. The quality of a parameter setting is indicated by the distances between a number of measurements, denoted as \(m_1, m_2, \dots , m_h\), and their theoretically optimal values, denoted as \(m^*_1, m^*_2, \dots , m^*_h\). Usually, dozens of measurements are considered at each operation point. The lower the distance is, the better the engine parameters are. It is worth mentioning that the engine parameters remain unchanged at different operation points.
Modelbased engine calibration with metaheuristics
The engine calibration process, illustrated in Fig. 1, can be formalised as an optimisation problem given a computational engine model, which simulates the model and calculates the measurements given a parameter setting and an operation point. Applying blackbox optimisation algorithms, such as metaheuristics, to automate the engine calibration process is straightforward. [28] reviewed the applications of metaheuristics in car engine calibration published till 2013. In the reviewed work [28], multiobjective evolutionary algorithms have often been applied to optimise at most six objectives at the same time [15, 21, 26], while singleobjective algorithms, such as particle swarm optimisation (PSO), have also been used [25, 32]. More recently, Ma et al. [22, 23] proposed a multiobjective optimisation approach to automate the gasoline direct injection engine calibration process, in which five engine parameters were adjusted under four operation conditions and three measurements were considered for each operation condition.
Multi and manyobjective optimisation
Various multiobjective evolutionary algorithms (MOEAs) and manyobjective evolutionary algorithms (MaOEAs) have been proposed to deal with multi and manyobjective optimisation problems [12, 24], mainly categorised into three groups: (i) Pareto dominancebased, (ii) decompositionbased, and (iii) indicatorbased. Pareto dominancebased MOEAs aim to use Pareto dominance mechanisms to select nondominated solutions and apply some additional strategies to improve the diversity of solutions. The nondominated sorting genetic algorithm II (NSGAII) [10] is one typical MOEA, and other examples include the Pareto envelopebased selection algorithm II (PESAII) [5] and the improved version of the Strength Pareto Evolutionary Algorithm (SPEA2) [36]. Although Pareto dominancebased MOEAs have good capability, the performance can be worsen rapidly with the increased number of objectives due to the dominance resistance phenomenon [17]. To overcome this issue, relaxed dominancebased MOEAs are proposed which aim to design new dominance relations, such as \(\epsilon \)dominance [16] and GrEA [33]. Examples of decompositionbased MOEAs/MaOEAs include the multiobjective evolutionary algorithm based on decomposition (MOEA/D) [34], the nondominated sorting genetic algorithm III (NSGAIII) [9], and the reference vectorguided evolutionary algorithm (RVEA) [2]. These algorithms tend to decompose the original problem into some simpler multiobjective or singleobjective optimisation problems. The twoarchive algorithm (Two_Arch2) [31], the indicatorbased evolutionary algorithm (IBEA) [35], and the fast hypervolumebased evolutionary algorithm (HypE) [1] are popular indicatorbased MOEAs/MaOEAs.
A realworld aeroengine calibration problem
We consider a real aeroengine model parameterised by 27 parameters which need to be adjusted at 33 operation points. The considered engine model is illustrated in Fig. 2.
Twentyseven engine parameters A total number of 27 engine parameters are adjusted, such as compressor pressure ratio, compressor efficiency, fan inner/outer pressure ratio, high/lowpressure turbine efficiency, bypass ratio, and heattransfer coefficient. Table 1 provides the lower and upper bounds of those parameters.
Fifteen measurements per operation point At each operation point, the computational model outputs 15 measurements of the considered real aeroengine, described in Table 2. These measurements describe the environmental condition, the working status, and the performance of the engine. The numbers between brackets in Fig. 2 correspond to the indices of measurements described in Table 2. Their desired values (called “targets” in this paper) for an engine of optimal status vary at different operation points. The targets of 33 operation points are available in our real problem.
Thirtythree operation points The measurements at 33 different operation points are provided. These operation points are carefully selected by a human engineer to represent various states of an aircraft.
Computational engine model A computational engine model (a Windows executable file with extension .exe) is available. It takes a parameter setting \(\varvec{x}\) and an operation point s as two inputs. During a simulation, several CFD calculations are performed inside the computational model to simulate the dynamics. If the CFD simulation does not converge, then the input setting is labelled as “infeasible” and the simulated measurement values are considered “unbelievable”; otherwise, the input setting is labelled as “feasible” and the simulated measurements \(m_1, m_2, \dots , m_h\) are considered “believable”. The label and simulated measurements are returned, as shown in Algorithm 1. Each simulation consumes approximately 0.2 s. Note that this is not the case of expensive optimisation problem, but the realtime optimisation efficiency is very important in this realworld application.
Evaluation criterion Given a number of operation points \(s_j\), \(j \in \{1,\dots ,33\}\), the quality of an engine parameter setting \(\varvec{x}\) is often evaluated with the rootmeansquare error (RMSE) computed as follows:
where \(m^*_{i,j}\) refers to the target value of the \(i^{th}\) measurement of the optimal engine at operation point \(s_j\) for any \(i \in \{1,\dots ,15\}\) and \(j \in \{1,\dots ,33\}\). The error rate of the \(i^{th}\) measurement at the \(j^{th}\) operation point is calculated as the normalised distance between the simulated value and its target value:
In our realworld scenario, an engine parameter setting \(\varvec{x}\) with \(RMSE(\varvec{x})\) lower than \(0.5\%\) is considered acceptable. For a better performance, an \(RMSE(\varvec{x})\) lower than \(0.3\%\) is favourable. To achieve a such quality, it takes on average 3 h manual work (as illustrated in Fig. 1) by an experienced human engineer.
Manyobjective evolutionary optimisation for aeroengine calibration
fSDE: fast MaOEA with shiftbased density estimation
In the field of multi/manyobjective optimisation, a performance indicator refers to the measure used to evaluate the quality of a solution set provided by an optimiser (e.g., an MOEA) [19]. The quality of a solution set can be described by the extent of its approximation of the Pareto front, in terms of four aspects [19], namely convergence, spread, uniformity, and cardinality. Similarly, indicators can also be applied to approximate the contribution of each individual (i.e., solution) in a population, and then used as a criterion to differentiate the individuals.
Indicatorbased MaOEAs have a significant advantage. They can avoid low selection pressure towards Pareto front via nondominated sorting thanks to a high proportion of nondominated solutions [17]. The studies of [17, 19] showed that generally the performance of an indicatorbased MaOEA strongly depends on the selection of appropriate indicators and strategies to use them. On one hand, different indicators have a different bias towards the Pareto front. For instance, the work of [18, 31] showed that IBEA using \(I_{\epsilon }\) indicator may make the population converge to subregions of the Pareto front with bad diversity. On the other hand, strategies using indicators imply large computational consumption and may lead to inefficiency of algorithms [1].
Due to the large number of objectives in this aeroengine calibration problem and the requirement of realtime efficiency in realworld application, we intend to design an indicatorbased MaOEA to achieve better performance with acceptable time cost.
Shiftbased density estimation (SDE)
Li et al. [20] showed that the shiftbased density estimation (SDE) is able to detect individuals with good convergence and diversity. Therefore, SDE [20] is chosen as the indicator to guide search processes as finding more diverse parameter settings of high quality is favourable in our case. The shifted density \(D(p,{\mathcal {P}})\) of an individual p corresponding to its belonging population \({\mathcal {P}}\) is formalised as follows [20]:
where m is the number of objectives. \(q_i\) and \(p_i\) (\(i \in \{1,2,...,m\}\)) denote the ith objective value of individuals q and p, respectively. The value \(D(p,{\mathcal {P}})\) is used as the fitness of individual p, thus \(fitness_{p} = D(p,{\mathcal {P}})\).
Framework of fSDE: less is more
SDE has been integrated into NSGAII, SPEA2, and PESAII, and shown to improve the performance of the aforementioned algorithms [20]. However, the aforementioned algorithms are complex and the required operations, such as nondominate sorting, are timeconsuming; while in our scenario, realtime efficiency is crucial. Therefore, we proposed a simple MaOEA framework with SDE, called fSDE, presented in Algorithm 2. First, N solutions are randomly generated as the initial population \({\mathcal {P}}\). Then, at every generation, a binary tournament selection strategy is processed to generate a mating pool B of size N based on the fitness of individuals in \({\mathcal {P}}\). After reproducing N offspring, denoted as O, through simulated binary crossover [7] and polynomial mutation [8], an environmental selection is processed to delete the N worst solutions from \({\mathcal {P}}\bigcup {\mathcal {O}}\) and keep the remained N solutions as the new population. The above steps are repeated until termination criteria are reached. Considering a population of N individuals for a multiobjective optimisation problem with m objectives, calculating the fitness values (i.e., shifted density values) costs \(O(mN^2)\). The deletion step in the environmental selection costs \(O(N\log {N})\). Therefore, the total complexity is \(O(mN^2)\).
Efficiency verification of fSDE on benchmark test suite
To verify the effectiveness of our proposed fSDE, we compare it to stateoftheart many and multiobjective evolutionary algorithms on a commonly used benchmark function set, MaF [3]. The baseline algorithms considered in our comparison study are MOEA/D [34], NSGAIII [9], Two_Arch2 [31], and IBEA [35], taken from the MATLABbased MOEA platform, PlatEMOv2.5 [30]. The proposed fSDE is also implemented in MATLAB and integrated into PlatEMOv2.5. MaF contains 15 test functions, designed for manyobjective optimisation aiming to represent various realworld scenarios with many different shapes of Pareto fronts [3]. Table 3 shows the number of objectives considered in this comparison study, the corresponding number of reference vectors, and the corresponding number of function evaluations [29], determined by Das and Dennis’s systematic approach [6, 9]. For all algorithms, its population size is set as the number of reference points.
The hypervolume (HV) is adopted as the performance metric. Normalised objective values are used to compute HV values, and then, the reference point \((1.1,1.1,\dots ,1.1)\) is used for computing HV. On each benchmark function of MaF, 30 independent optimisation trials have been performed by each algorithm. The averaged HV values, standard deviations, and the results of Wilcoxon rank sum test [4] in solving MaF with 5, 10, and 20 objectives are reported in Tables 4, 5, and 6, respectively.
According to Table 4 (5 objectives), Two_Arch2 performs better than fSDE and its pairwise win/tie/loss counts against fSDE are 8/4/3. fSDE outperforms MOEA/D and NSGAIII in terms of HV values. In addition, the performance of IBEA and NSGAIII is similar to that of fSDE. In solving MaF test suite with ten objectives (cf. Table 5), fSDE has the best overall performance in terms of convergence and diversity among the compared MaOEAs. fSDE achieves significantly better HV values than MOEA/D in 12 out of 15 functions. When solving MaF with 20 objectives (cf. Table 6), NSGAIII obtains better HV values than fSDE, while fSDE achieves better quality (HV) than MOEA/D. Moreover, IBEA and Two_Arch2 have similar performance compared to fSDE. To summarise, fSDE achieves superior or competitive performance on MaF compared to the considered baseline algorithms.
fSDE for aeroengine calibration
We model the aeroengine calibration process as a manyobjective optimisation problem, aiming at minimising simultaneously all the error rates, Eq. (2). To do so, two naive but effective dimensionality reduction methods (“Naive but effective dimensionality reduction”) and dimensionality reduction via principal component analysis (“Failure of dimensionality reduction via principal component analysis”) have been performed.
Naive but effective dimensionality reduction
Two naive dimensionality reduction methods have been considered: (i) modelling \(\frac{1}{15}\sum _{i=1}^{15} e_{i,j}(\varvec{x})\) with \(j \in \{1,\dots ,33\}\) as 33 objectives; and (ii) modelling \(\frac{1}{33}\sum _{j=1}^{33} e_{i,j}(\varvec{x})\) with \(i \in \{1,\dots ,15\}\) as 15 objectives. As a baseline, we also (iii) model all the error rates as \(33*15\) objectives. Twenty independent optimisation trials of fSDE with 20,000 simulation calls to the computational model as optimisation budget are applied to each of the above three manyobjective optimisation problems, denoted as fSDE33, fSDE15, and fSDE495, respectively.
Quality assessment in objectives Figure 3 illustrates the averaged RMSE, calculated with Eq. (1), over 20 independent trials with respect to the simulation number (left) and time in seconds (right). No obvious difference has been observed among the three cases at the early optimisation stage. However, along with the optimisation time, fSDE15 (solid black curve) converges slightly better than fSDE33 and fSDE495. Figure 4 illustrates the lowest error rate among the 495 error rates among the 20 trials, named as minmax error. Lower minmax error implies overall better parameter setting. According to Figs. 3 and 4, although fSDE33 achieves similar RMSE as the other two cases, its minmax error is higher after 5, 000 model simulations. The experimental result is coherent with an engineering perspective, and thus, the values of an identical measurement at various operation points is highly correlated and can be aggregated to a single value, as in case (i), thus fSDE15.
Quality assessment in decisions Four aeroengine parameter settings with RMSE lower than \(0.5\%\) found by fSDE15, denoted as \(\varvec{x}_a\), \(\varvec{x}_b\), \(\varvec{x}_c\), and \(\varvec{x}_d\), are illustrated in Table 7. For reference, a known optimum parameter setting of this problem, denoted as \(\varvec{x}^*\), is given on the second column. On most of the coordinates, the illustrated parameter settings are very closed to the optimum, except for the parameters at coordinates 15–17. Those parameters at coordinates 15–17 are the volume of cooling flow of the highpressure turbine guide vane, the highpressure turbine rotor, the lowpressure turbine guide vane, and the lowpressure turbine rotor. The model is less sensitive to those four parameters.
Quality assessment in efficiency As mentioned previously, in this calibration problem, an experienced human engineer needs approximately 3 h to find a parameter setting with an RMSE value lower than \(0.3\%\). With fSDE, it takes less than 15 min to achieve a such RMSE value and 22 min to achieve an RMSE value of \(0.2\%\). All the experiments in this paper were ran on an i78700 Intel processor desktop with 8 cores.
Failure of dimensionality reduction via principal component analysis
To reduce the dimensionality of objectives, principal component analysis (PCA) has been performed. First, we sample 20,000 parameter settings uniformly at random in the legal ranges provided in Table 1 and simulate them at the given operation points. Among the 20,000 samples, only 13,331 settings are feasible. PCA is performed on the 15 averaged measurements over operation points of feasible samples, according to the results of which the number of objectives is reduced to 7. Again, fSDE is applied to optimise the resulted 7objective problem, denoted as fSDE7. However, after dimensionality reduction via PCA, the percentage of infeasible settings sampled by fSDE (left of Fig. 5) increases along with the increased generation number, while without dimension reduction, the percentage of infeasible settings sampled by fSDE15 (cf. Sect. 4.3.1) decreases, as shown in the right of Fig. 5. Due to the large number of sampled infeasible parameter settings, the fSDE fails and does not converge given more optimisation time. The dimensionality reduction via PCA loses the implicit information of parameter feasibility and infeasibility.
Comparison with baselines
As a supplementary verification, IBEA, MOEA/D, NSGAIII, and Two_Arch2 with the second dimensionality reduction method, i.e., case (i), have also been applied given the same number of simulations and the results are illustrated in Figs. 3 and 4. Our proposed fSDE outperforms NSGAIII and Two_Arch2 in terms of solutions quality (RMSE) and outperforms MOEA/D in terms of realtime performance (right of Fig. 3) on our real aeroengine calibration problem.
Discussion
When modelling our aeroengine calibration as a manyobjective optimisation problem, the usage of sophisticated dimensionality reduction method should be reconsidered. On the contrary, naive dimensionality reduction methods meet better the needs in engineering and lead to better results. Manyobjective evolutionary algorithms are able to find diverse solutions of high quality. However, their computational cost is sometimes relatively high if an evaluation of solution is not feasible. As in our case, an evaluation takes only around 0.2 s and the optimisation efficiency is a more important indicator to be considered when selecting an optimisation algorithm. Our proposed fSDE with a simple framework outperforms the baseline algorithms in terms of realtime performance on the aeroengine calibration.
Singleobjective evolutionary optimisation for aeroengine calibration
For a further verification of our approach, we model the calibration process as a singleobjective optimisation problem with a linear aggregation of the distance between the target values and the simulated values of measurements given a setting. The fitness function is designed as follows:
where \(RMSE(\cdot )\) is defined in Eq. (4). A huge negative value is returned as fitness in case of invalid calculation or constraint violation in CFD calculations.
Failure of PSO, DE, and CMAES Although classic singleobjective evolutionary algorithms can be applied directly to a such problem, two issues arise: (i) sampling infeasible settings leads to slow convergence due to the time consumed to determine the feasibility of sampled settings, and (ii) resamplings due to sampled outofrange settings lead to divergent values. The optimisation results by a classic PSO [13], a differential evolution (DE) [27], and a covariance matrix adaptation evolution strategy (CMAES) [11] shown in Figs. 6 and 7 and Table 8 illustrate examples of the above two issues. Therefore, particular operations should be taken for tackling the above issue.
The classic PSO and DE are implemented by us. The PSO uses \(\lambda = 90\), \(\omega =0.5\), and \(C_1=C_2=0.5\), and the DE uses \(\lambda = 90\), \(C_r=0.3\) and \(F=1\). The program of CMAES is taken from https://github.com/AlexanderFabisch/CMAESpp, and uses \(\mu =\lambda =13\), while the other parameters use the default setting. Each curve in Figs. 6 and 7 is an averaged result over 20 independent optimisation trials with 20,000 model simulations as budget.
Selfadaptive particle swarm optimisation (saPSO)
In the classic PSO, the location of particles (\(\varvec{x}\)) is updated by the following equations:
where \(\varvec{x}_i\) is the \(i^{th}\) individual, and \(r_1\) and \(r_2\) are two random vectors \(\in (0,1)^d\), where d is the dimension. Each coordinate of the velocity \(v_i\) is the sum of three items: inertia (\(\omega * v_{i}\)), the vector with random weight pointing to the global best location of all particles found during optimisation (\(C_1 * r_1 * (gbest  \mathbf{{x}}_{i})\)) and the vector with random weight pointing to the best location of this particle found during optimisation (\(C_2 * r_2 * (pbest_{i} \mathbf{{x}}_{i})\)).
To avoid the change of sampling distribution, the way to handle the outofrange or infeasible solutions is resampling. However, when a particle is close to the boundary of feasible area in solution space and the inertia (\(\omega *v_{i}\)) points out to the boundary, the updated location (\(\mathbf{{x}}_i\)) is likely to be infeasible. Larger the inertia is, lower the possibility of sampling a feasible solution is. As a consequence, more computational resource will be consumed by resampling. In PSO, the waste of resource on resampling leads to a significant reduction of final solution’s quality given limited optimisation budget.
We propose a selfadaptive PSO (saPSO) to handle this situation. In saPSO, for each resampling, \(\omega \) is assigned as \(\omega ^2\). As \(\omega \in (0,1)\), the inertia is attractively reduced during resampling. Therefore, the possibility that the generated location is feasible is increasing. Algorithm 3 describes the process of our saPSO. The saPSO with the same parameters as PSO and \(\omega =0.9\) is performed on the engine calibration using the fitness defined in Eq. (4). Experimental results are shown in Figs. 6 and 7, as well as Table 8. It is observed that the saPSO converges as fast as CMAES, but does not lead to solution value far from the optimum on the \(18^{th}\) coordinate as the CMAES, PSO, and DE do. The extremely poor values found by CMAES, PSO, and DE on the 18th coordinate can be explained by the sampled outofrange or infeasible parameter settings. When an outofrange parameter setting is sampled, CMAES, PSO, and DE will resample a setting uniformly at random within the ranges given in Table 1. Although the particular design of saPSO successfully reduces the probability of sampling outofrange or infeasible parameter settings, its final recommended parameter setting is not as good as the ones recommended by fSDE given the same amount of model simulations (cf. Fig. 8).
Discussion
Modelling the aeroengine calibration process as a singleobjective optimisation problem is straightforward. However, due to the characteristics of our realworld calibration problem, the singleobjective evolutionary algorithms are not efficient due to the sampling of outofrange or infeasible parameter settings. Although we have proposed a simple selfadaptive PSO particularly to tackle this issue and obtained better performance comparing to some classic singleobjective evolutionary algorithms, the efficiency of the selfadaptive PSO is still lower than the fSDE (cf. Fig. 8). This observation can be explained by the mechanics of fSDE. The crossover and mutation operators of fSDE will not lead to outofrange samples.
Conclusion
In this paper, we consider a real aeroengine calibration problem with 27 parameters to be calibrated, 15 measurements at each operation point as quality criteria, and their values at 33 operation points as calibration data. The calibration process is modelled as a blackbox optimisation problem aiming at minimising the distance between the simulated measurement values using a computational engine model and their calibration data. Both many and singleobjective evolutionary algorithms have been applied to calibrate the parameters. A fast manyobjective evolutionary optimisation algorithm with shiftbased density estimation, called fSDE, and a novel singleobjective evolutionary algorithm, called saPSO, have been proposed and compared to several classic and stateoftheart algorithms. The performance of fSDE has been assessed in terms of objective values, recommended decisions, and efficiency. Without parallelising the calculation or evaluation of population, the proposed fSDE consumes ten times less time to find diverse parameter settings of high quality, compared to the time needed to find one single setting by an experienced human engineer using the traditional manual calibration process. fSDE not only provides diverse solutions but also finds solutions of highquality faster, compared to some stateofthearts, thank to its simple operations.
Besides problem solving, our work also contributes in various ways. When finding different engine parameter settings with similar performance, human engineers may be assisted to diagnose the computational engine model and the measurements, and inspired to discover novel engine models. As a future work, we are interested in training a predictor to determine the feasibility of a sampled parameter setting to further reduce the required number of model simulations and improve the optimisation efficiency.
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Funding
This work was supported by the AECC, the National Natural Science Foundation of China (Grant Nos. 61906083, 61976111), the Guangdong Provincial Key Laboratory (Grant No. 2020B121201001), the Program for Guangdong Introducing Innovative and Entrepreneurial Teams (Grant No. 2017ZT07X386), the Shenzhen Science and Technology Program (Grant No. KQTD2016112514355531), and the Science and Technology Innovation Committee Foundation of Shenzhen (Grant No. JCYJ20190809121403553).
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Liu, J., Zhang, Q., Pei, J. et al. fSDE: efficient evolutionary optimisation for manyobjective aeroengine calibration. Complex Intell. Syst. 8, 2731–2747 (2022). https://doi.org/10.1007/s40747021003741
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DOI: https://doi.org/10.1007/s40747021003741
Keywords
 Engine calibration
 Manyobjective optimisation
 Multiobjective optimisation
 Constrained optimisation
 Evolutionary algorithm