ARTICLE IN PRESS. Applied Energy xxx (2009) xxx xxx. Contents lists available at ScienceDirect. Applied Energy

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1 Applied Energy xxx (9) xxx xxx Contents lists available at ScienceDirect Applied Energy journal homepage: GA-based design-point performance adaptation and its comparison with ICM-based approach Y.G. Li *, P. Pilidis School of Engineering, Cranfield University, Bedford MK AL, United Kingdom article info abstract Article history: Received 8 February 9 Received in revised form 7 April 9 Accepted 9 May 9 Available online xxxx Keywords: Gas turbine Engine Performance adaptation Performance matching Design-point performance simulation Influence Coefficient Matrix (ICM) Genetic Algorithm (GA) Accurate performance simulation and estimation of gas turbine engines is very useful for gas turbine manufacturers and users alike and such a simulation normally starts from its design-point. When some of the engine component parameters for an existing engine are not available, they must be estimated in order that the performance analysis can be started. Therefore, the simulated design-point performance of an engine may be slightly different from its actual performance. In this paper, a Genetic Algorithm (GA) based non-linear gas turbine design-point performance adaptation approach has been presented to best estimate the unknown component parameters and match available design-point engine performance. In the approach, the component parameters may be compressor pressure ratios and efficiencies, turbine entry temperature, turbine efficiencies, engine mass flow rate, cooling flows, by-pass ratio, etc. The engine performance parameters may be thrust and SFC for aero engines, shaft power and thermal efficiency for industrial engines, gas path pressures and temperatures, etc. To select the most appropriate to-be-adapted component parameters, a sensitivity analysis is used to analyze the sensitivity of all potential component parameters against the engine performance parameters. The adaptation approach has been applied to an industrial gas turbine engine to test the effectiveness of the approach. The approach has also been compared with a non-linear Influence Coefficient Matrix (ICM) based adaptation method and the advantages and disadvantages of the two adaptation methods have been compared with each other. The application shows that the sensitivity analysis is very useful in the selection of the to-beadapted component parameters and the GA-based adaptation approach is able to produce good quality engine models at design-point. Compared with the non-linear ICM-based method, the GA-based performance adaptation method is more robust but slower in computation and relatively less accurate. Ó 9 Elsevier Ltd. All rights reserved.. Introduction * Corresponding author. Tel.: + 757; fax: address: i.y.li@cranfield.ac.uk (Y.G. Li). Performance of gas turbine engines is different for different types of engines, and even varies from engine to engine in the same fleet due to manufacturing and assembly tolerance. Thermodynamic modelling techniques and computer software for gas turbine performance simulation has become more and more sophisticated in building high quality performance models. However, it is still difficult to build accurate performance models for gas turbine engines even at design-point without enough accurate engine component performance information, such as air flow rate, compressor pressure ratios and isentropic efficiencies, turbine entry temperature, turbine efficiencies, cooling flows, etc. Much of this engine performance information is OEM s proprietary information so that they can not be accessed easily and also can not be measured directly. It is therefore very difficult for any third parties to build high quality performance models that are sometimes crucially important. Even with OEM s engine performance information, accurate performance models, which are able to distinguish the performance difference among the same fleet engines, are highly desirable in some applications such as condition monitoring and diagnostics. Gas turbine design-point performance models can be created and improved to match available or measured engine performance by manually selecting and adjusting design-point component parameters. However, this task may become very difficult even for engineers with good performance knowledge and experience when engines are complex in configuration. In other words, there may be too many design-point component parameters to estimate in order to allow a large number of engine measurable performance parameters to be matched. Gas turbine design-point performance status estimation or adaptation methods have been explored by some researchers. Roth et al. introduced an optimization concept [] for engine cycle model matching and a minimum variance estimator algorithm [] for performance matching of a turbofan engine. Li et al. introduced an Influence Coefficient Matrix 6-69/$ - see front matter Ó 9 Elsevier Ltd. All rights reserved. doi:.6/j.apenergy.9.5. (9), doi:.6/j.apenergy.9.5.

2 Y.G. Li, P. Pilidis / Applied Energy xxx (9) xxx xxx Nomenclature Symbols a weighting factor ACM Adaptation Coefficient Matrix GA Genetic Algorithm H Influence Coefficient Matrix h( ) vector-valued function ICM Influence Coefficient Matrix M number of target performance parameters m a engine inlet air flow rate (kg/s) m c /m c cooling flow rate to turbines and N number of to-be-adapted component parameters OF Objective Function P total pressure (atm) PR pressure ratio SFC specific fuel consumption (mg/n s) T total temperature (K) TET Turbine entry temperature (K) Ts static temperature (K) ~x actual component parameter vector ^~x estimated to-be-adapted component parameter vector ~z actual target performance parameter vector ^~z simulated target performance parameter vector e adaptation error D deviation DP b Burner relative pressure loss (%) P summation r convergence threshold g component isentropic efficiency Subscripts b Burner c Compressor c Compressor t turbine /compressor turbine t turbine /power turbine baseline point 6 Compressor exit compressor turbine exit nozzle exit cooling flow from Compressor exit (ICM) based adaptation method [] for gas turbine design-point performance adaptation and successfully applied the method to the test data of an industrial gas turbine engine. Initial study of the GA-based adaptation method was provided by Li []. In this paper, a further investigation of the GA-based non-linear design-point performance adaptation method and its comparison with a ICM-based adaptation method [] is carried out and is applied to the design-point performance status estimation of an industrial gas turbine engine GE LM5+ running at Manx Electricity Authority (MEA), UK. The advantages and disadvantages of both adaptation methods are discussed.. Methodology for adaptation Design-point performance adaptation is an inverse mathematical problem. In other words, the information of the dependent variables in a system is used to estimate the independent variables in the system. In design-point performance adaptation process, two types of adaptation parameters are defined and used: To-be-adapted component parameters They are also called independent component parameters or gas turbine design-point component parameters, such as air flow rate, compressor pressure ratios and isentropic efficiencies, turbine entry temperature, turbine isentropic efficiencies, and cooling flows. These parameters are independent from each other and determine the engine performance at design-point. Target performance parameters They are also called dependent performance parameters, measurable performance parameters, such as thrust or shaft power, SFC or thermal efficiency, and gas path pressures and temperatures. The quantity of these parameters is determined by the quantity of the independent component parameters. The thermodynamic relationship between gas turbine to-beadapted component parameters and target performance parameters of a gas turbine engine can be expressed as: ~z ¼ h ðþ ~x ðþ where ~z R M is the target performance parameter vector and M the number of the target performance parameters, ~x R N is the to-beadapted component parameter vector and N the number of the to-be-adapted component parameters and h() is a vector-valued function representing engine performance behaviour... GA-based performance adaptation method The design-point gas turbine performance adaptation can be regarded as an optimization problem whose idea is shown in Fig.. At certain ambient and operating condition and for an engine with certain design-point component parameters ^~x; simulated engine performance ^~z from an engine modelis compared with the target performance ~z. The difference ^~z ~z is used by an optimization algorithm to update the estimation of the to-be-adapted engine component parameters ^~x whose initial value should be given to the performance model. When measured performance parameters ~z at design operating condition is set as a target of the adaptation, any potential adaptation solution ^~x is evaluated by means of an Objective Function (OF) defined as: Real Engine Target performance z GA Fitness=Max? Stop Simulated performance ẑ Yes Engine Model No Estimated to-be-adapted parameters xˆ Fig.. GA-based design-point adaptation method. (9), doi:.6/j.apenergy.9.5.

3 Y.G. Li, P. Pilidis / Applied Energy xxx (9) xxx xxx OF ¼ XM a i ^z i z i M i¼ z i where z i are the actual values of the target performance parameters and ^z i are the corresponding simulated performance parameters from the engine performance model. a i are the weighting factors taking into account the uncertainty and relative importance of the target performance parameters. The uncertainty of the measured parameters is not considered in this study and all the measurements are regarded as equally important, therefore the weighting factors are set as a i =. (i= M) for simplicity. The adaptation error for each target performance parameter is defined by the relative difference between the predicted and actual engine target performance parameter and expressed as: e i ¼ ^z i z i z i % ðþ where e i is the adaptation error for i th target performance parameter. In this study, a Genetic Algorithm (GA) is used as the optimization algorithm for the searching of the best set of the to-beadapted component parameters due to its superior advantages to conventional optimization methods. Genetic Algorithms (GA) follow the idea of Darwin s natural evolution and is one of the optimization searching techniques. A real coded GA is used in the study to ensure accurate representation of individuals within a GA population. Unlike conventional optimization methods, GA offers several unique features. For example, GA combines elements of directed and stochastic search, it is a global search to avoid getting stuck in local optima, and no derivatives are required so that non-smooth functions can be optimized. Due to these features, GA is chosen for the design-point performance adaptation although it is generally known to be computationally more expensive than conventional optimization methods. The quality of individual solutions within the GA population is assessed by its Fitness defined as: Fitness ¼ ðþ þ OF where the Objective Function (OF) defined in Eq. () represents the difference between predicted and actual measurements. Thus, a value of Fitness approaching indicates a good solution while a value of Fitness approaching indicates a poor one. A Genetic Algorithm normally uses three GA operators in its searching process: reproduction (or selection), crossover and mutation. Along with the three GA operators, an elitism operation [5] is used. With this operation, the worse strings in GA population of a generation are replaced with better strings generated by the three GA operations and a fitter GA population is produced in the next generation. Therefore, it is ensured that the average Fitness of the whole population will improve from one generation to the next. The results of crossovers and mutations in each generation of a GA search are strongly affected by the probabilities assigned to these operators. The accuracy of the results and the speed of the search depend on the number of GA generations used and the size of the GA population. A higher number of GA generations with bigger size of GA population are beneficial to better solutions with the drawback of higher computational costs. A necessary compromise must be carefully made after several trial runs. The best values of the GA parameters are problem specific and the typical values used in this study are shown in Table where different values of the GA parameters are used for initial and final adaptation calculations. Small value of GA parameters helps GA to find correct searching space in the trial calculations with relatively ðþ Table GA parameter setting. No. Parameters Value Initial calculation Final calculation Population size No. of generations Probability of cross-over..6 Probability of mutation..7 high computation speed while large value of GA parameters tend to provide accurate solutions in final GA calculations. Upper and lower bound for each of the to-be-adapted parameters should be specified properly in order that the searching space covers the best potential solutions and good quality solutions can be obtained. Trials and errors are normally used to adjust the upper and lower bounds of the to-be-adapted component parameters in initial trial calculations. Due to the fact that the focus of this paper is to use the Genetic Algorithm as an optimization tool rather than the development of the technique, more detailed information about Genetic Algorithms are not described in the paper and the interested readers may read relevant books such as [5]... Selection of adaptation parameters and sensitivity analysis Proper selection of the to-be-adapted component parameters is crucial to the success of the adaptation. Some important factors should be taken into account in the selection of those to-beadapted component parameters and they are as follows: The selected to-be-adapted component parameters should have strong functional relationship with the target performance parameters. Otherwise, they do not provide sufficient searching space in the adaptation and do not contribute much to better adaptation results No limitation is applied to the number of the to-be-adapted component parameters but the solutions of the adaptation may be different if different parameters are selected. In addition, a larger number of to-be-adapted component parameters are likely to provide larger searching space and provide better adaptation solutions. The upper and lower bounds of the to-be-adapted component parameters also determine the domain of the search space. Therefore, proper selection of the bounds is very important in the adaptation and several trials may be required to ensure that the best potential solutions are included in the searching space. A sensitivity analysis where the deviations of the target performance parameters due to seeded unit deviation of individual tobe-adapted component parameters is used to identify the most appropriate to-be-adapted component parameters. For the to-beadapted component parameters to be chosen effectively, part of the target performance parameters must be sensitive enough to each of the chosen to-be-adapted component parameters. In other words, the percentage deviation response of the target performance parameters should be ideally similar to or even larger than the seeded percentage change of the chosen to-be-adapted component parameters. The demonstration of the sensitivity analysis is shown later in the paper... ICM-based adaptation method An Influence Coefficient Matrix (ICM) based design-point performance method was published by Li et al. [] for the same perfor- (9), doi:.6/j.apenergy.9.5.

4 Y.G. Li, P. Pilidis / Applied Energy xxx (9) xxx xxx mance adaptation purpose. To assist a comparison between the GA-based adaptation method and the ICM-based adaptation method, a brief description of the ICM-based adaptation method is given as follows. An initial engine model presented with Eq. () may be expanded in a Taylor series and a linearized relationship between the deviation of the to-be-adapted component parameters and the deviation of corresponding target performance parameters of a gas turbine around the baseline point of the gas turbine performance model can be expressed as: D~z ¼ H D~x where D~z ¼ ~z ^~z; D~x ¼ ~x ^~x; H is Influence Coefficient Matrix (ICM), ^ represents baseline condition of the gas turbine performance model. An engine performance model may be built based on initial guess of the to-be-adapted component parameters and such guess is prone to errors. Correspondingly, initially predicted target performance parameters (^~z) may be different from their actual values (~z) and such a difference is the prediction error of the performance parameters relative to their target represented by D~z. Therefore, the correction of the to-be-adapted component parameters D~x can be predicted by inverting the Influence Coefficient Matrix (ICM) H to a Adaptation Coefficient Matrix (ACM) H leading to Eq. (6) when N=M: D~x ¼ H D~z If N>M, Eq. (5) is under-determined and a pseudo-inverse is defined as: H # ¼ H T ðhh T Þ If N<M, Eq. (5) is over-determined and the pseudo-inverse is defined as: H # ¼ðH T HÞ H T The solution resulting from the pseudo-inverse D~x ¼ H # D~z is best in a least-squares sense. The linear prediction of D~x may be a good estimate of the correction of the to-be-adapted component parameters when the thermodynamic behaviour of the engine around the baseline condition is close to linear. However, it is very likely that the predicted correction of the to-be-adapted component parameters is far from Prediction error of target performance parameters Δz rd Iteration nd Iteration Predicted Solution Exact Solution ð5þ ð6þ ð7þ ð8þ ð9þ accurate when the nonlinearity of the relationship between the tobe-adapted component parameters and the target performance parameters is significant. To improve the prediction accuracy a Newton Raphson iterative method is used where the linear prediction is applied iteratively until a converged solution is obtained and this process is shown in Fig.. The convergence of the iterative adaptation calculation process shown in Fig. is declared when the predicted performance is very close to the real target performance. This criterion is shown in Eq. () where the root mean square (RMS) of the difference between the predicted and target performance parameters is smaller than r when convergence is declared: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P u M ^z i z t i i¼ z i RMS ¼ < r ðþ M where r is a very small number (. is chosen in this study).. Application and comparison.. LM5+ and performance simulation The gas turbine engine used in this research is a General Electric industrial LM5+ engine installed in Isle of Man, UK by MANX Electricity Authority (MEA). MEA has two LM5+ engines, each producing about 9 MW (dry mode), driving a combined cycle power plant with two Once-Through Steam Generators and a steam turbine producing 5 MW [6]. The LM5+ industrial gas turbine (Fig. ) is a two-shaft engine with one compressor, one burner, one compressor turbine and one power turbine. The available nominal performance parameters of the engines (dry mode) are as follows [7]: Exhaust gas flow rate: 8 kg/s Total pressure ratio:. Power output: 9 MW Thermal efficiency: 8.8% Actual gas path performance parameters of the engines are measured regularly in MEA and one set of measurements were selected and used in this study. The measurement chosen for the design-point performance adaptation was taken when an engine ran steadily at an operating point very close to a nominal operating condition regarded as an artificial design-point; such design-point might be different from the design-point set by engine manufacturer. Measurement bias is not considered and the measured parameters are regarded as the representative of the parameters at corresponding gas path stations. To reduce the influence of the measurement noise, multiple measurement samples were obtained and averaged to get averaged values of the measurements. Therefore, the values of the gas path parameters taken from the measurement can only be seen as an approximation of the true values of those parameters. The available gas path measurable parameters of the engine are shown in Table. Gas turbine design-point performance is simulated using software TURBOMATCH/PYTHIA [8,9] developed at the School of Engineering, Cranfield University. The configuration of the engine model used in this study is shown in Fig. where two compressors are used to represent the whole compressor of the engine. The cooling flows for the two turbines are taken from the exit of the two compressors. Baseline Correction of to-be-adapted component parameters Fig.. Non-linear ICM-based adaptation []. Δx.. Selection of adaptation parameters and sensitivity analysis Among the two types of adaptation parameters being used in the performance adaptation, the target performance parameters (9), doi:.6/j.apenergy.9.5.

5 Y.G. Li, P. Pilidis / Applied Energy xxx (9) xxx xxx 5 Fig.. GE LM5+ gas turbine engine [7]. Table Gas path measurable parameters. No. Symbol Unit Meaning T 6 K Total temperature at compressor exit P 6 atm Total pressure at compressor exit T K Total temperature of cooling flow from 7th stage of compressor P atm Total pressure of cooling flow from 7th stage of compressor 5 T K Total temperature at compressor turbine exit 6 P atm Total pressure at compressor turbine exit 7 Ts K Static temperature at exhaust 8 m f kg/s Fuel flow rate Compressor Compressor Burner Turbine Turbine Nozzle Fig.. LM5+ engine model. are shown in Table and the potential to-be-adapted component parameters are shown in Table. The nominal values of the tobe-adapted component parameters may be available in an engine manual or the public domain but their actual values may be slightly different even for engines of the same type due to manufacturing and assembly tolerance and engine health conditions. Initial guess may be given to those whose values are not available. To choose the best set of the to-be-adapted component parameters, a sensitivity analysis showing the response of the target performance parameters to the to-be-adapted component parameters is shown in Figs. 5a and 5b. It can be seen that the target performance parameters respond differently to different to-be-adapted component parameters and the levels of the response can be significantly different. For example, T, P, Ts and m f respond strongly to the seeded deviation of TET, and all measurable parameters respond weakly to the seeded deviation of m c and DP b. The better selection of the to-be-adapted component parameters should be those that respond more sensitively to the seeded deviation of the to-be-adapted component parameters. Initial values of the to-be-adapted component parameters must be given to obtain a baseline engine model and to start the performance adaptation. Those initial values close to their actual values will ease the selection of upper and lower bounds of the to-beadapted component parameters in GA adaptation and achieve better adaptation solutions. Therefore, the initial selection of the tobe-adapted component parameters should be done manually in such a way that the predicted values of the target performance parameters are as close as possible to that of the actual performance parameters. Then the more accurate performance models can be obtained with the adaptation approaches... Adaptation with different selection of to-be-adapted component parameters For the same engine model and the same set of target measurable parameters, different selections of the to-be-adapted component Table To-be-adapted component parameters. No. All potential to-be-adapted parameters Unit Meaning Four sets (-selected) Set A Set B Set C Set D m a kg/s Engine air flow rate PR c Compressor pressure ratio g c % Compressor isentropic efficiency PR c Compressor pressure ratio 5 g c % Compressor isentropic efficiency 6 TET K Turbine Entry Temperature 7 g b % Burner combustion efficiency 8 g t % Turbine isentropic efficiency 9 g t % Turbine isentropic efficiency m c kg/s Cooling flow flow rate m c kg/s Cooling flow flow rate DP b % Burner pressure drop (9), doi:.6/j.apenergy.9.5.

6 6 Y.G. Li, P. Pilidis / Applied Energy xxx (9) xxx xxx ma PRc η PRc ηc TET T6 P6 T P T P Ts mf Fig. 5a. Sensitivity comparison (% seeded deviation to each of the to-be-adapted component parameters) ηb ηt ηt mc mc ΔPb T6 P6 T P T P Ts mf Fig. 5b. Sensitivity comparison (% seeded deviation to each of the to-be-adapted component parameters). parameters may result in different adapted engine models with different adaptation accuracies. To analyze the impact of such selections, four sets of to-beadapted component parameters are selected (Sets A, B, C and D) based on the information shown in Figs. 5a and 5b and they are listed in Table. Among the four sets, Set A includes all the potential to-be-adapted component parameters, Set B includes all the parameters that show significant sensitivity to most of the target performance parameters, Set C tries to use a minimum number of the to-be-adapted component parameters that are most sensitive to the target performance parameters and Set D includes all the parameters except two most sensitive to-be-adapted component parameters (TET and g t ). For the same target measurable parameters (Table ), the GAbased adaptation approach is applied to the first three sets of the to-be-adapted component parameters. The shifting of the to-beadapted parameters from their initial values is shown in Fig. 6 which shows that the adaptation provides different solutions of the to-be-adapted component parameters for the different sets of the to-be-adapted parameters. The adaptation accuracy resulted from the three sets of the to-be-adapted parameters is shown in Fig. 7 where although the adaptation solutions from all the three to-be-adapted component parameter sets have much improved accuracy compared with the initial errors, Set A provides the best adaptation result while Set C provides the worst result. In other words, different selection of the to-be-adapted component parameters results in different engine models with different adaptation accuracies. It can also be seen that the GA fitness can achieve as high as around.99 (Fig. 8) and the maximum adaptation error for individual target measurable gas path parameters can be as low as.5% in the result using Set A. In the GA-based adaptation method, there is no restriction in the selection of the to-be-adapted component parameters; the method can always provide a solution if properly used, showing Shifting (%) ma PRc ηc PRc ηc TET that this method is numerically very robust. However, the quality of the solutions is strongly affected by the selection of the to-beadapted component parameters and their corresponding upper and lower bounds. This is due to the fact that the adaptation searching space is determined by the number of the to-be-adapted component parameters and their upper and lower bounds. The difference in the quality of the adapted solutions is demonstrated in Figs. 7 and 8 where the individual adaptation errors and the GA Fitness are shown, respectively. The upper and lower bound of each of the to-be-adapted parameters should be specified properly in order that the searching space covers the best potential solutions and good quality solutions can be obtained. Initial bounds of the to-be-adapted parameters are chosen at ±% of and centred on the initial value of the to-be-adapted component parameters. Trials and errors are used to adjust the upper and lower bounds in order that the ηb ηt ηt Set A Set B Set C Fig. 6. Adaptation results from Sets A, B and C. mc mc ΔPb (9), doi:.6/j.apenergy.9.5.

7 Y.G. Li, P. Pilidis / Applied Energy xxx (9) xxx xxx 7 Adaptation Error (%) T6 P6 T P T P Ts mf Initial error Set A Set B Set C Fig. 7. Comparison of adaptation accuracy. GA Fitness Calculation Fig. 9. Comparison of GA fitness from repeated adaptation calculations..9.8 GA Fitness Set A Set B Set C Fig. 8. Comparison of fitness. Shift from Initial Value (%) ma ICM-based GA-based PRc ηc PRc ηc TET ηt ηt ηb mc mc dpb expected final solution of the to-be-adapted parameters would more or less settle between the upper and lower bounds. Due to the nature of the Genetic Algorithm, the GA-based adaptation method is much more time consuming than other numerical optimization or searching methods, such as the ICM-based method described in []. By using a personal computer with Pentium IV processor, it takes around 5 s for one generation and min 5 s for a whole calculation for a typical initial adaptation calculation setting (Table ). For a final adaptation calculation setting (Table ), it takes around s for one generation and 5 min and s for a whole final adaptation calculation. The GA searching method is also stochastic in nature. Therefore, the GA-based adaptation may provide slightly different solutions if the adaptation calculation with exactly the same adaptation setting is repeated. To ensure that the GA-based adaptation method provides reliable solutions, the adaptation calculation with the same adaptation setting and the same set of the to-be-adapted component parameters (Set A) were repeated times. A comparison of the obtained GA Fitness is shown in Fig. 9 where the majority of the Fitness are between.97 and.99, very close to each other with only one exception where the Fitness is around.9. This indicates that the GA-based adaptation is able to provide reliable solutions... Comparison with ICM-based adaptation method... Adaptation accuracy For the same set of the target performance parameters (Table ), both the GA-based and the ICM-based adaptation approaches - Fig.. Results of two adaptation methods using Set A. are applied to Set A of the to-be-adapted component parameters shown in Table. The corrections of the to-be-adapted parameters to their initial value obtained from the two adaptation methods are shown and compared in Fig. indicating that the two adaptation methods provide similar but slightly different results. The shifts of dp b seam to be quite different in percentage but the absolute value of the shifts are very small due to the small absolute value (baseline value dp b =.5) of the parameter and also it is one of the least sensitive to-be-adapted component parameters for the target performance parameters (Fig. 5b). The adaptation accuracy resulted from the two adaptation methods are shown and compared in Fig. where the adaptation errors for the individual target performance parameters obtained from the ICM-based adaptation method are almost invisible while the largest adaptation errors (for T 6, T and P ) obtained from the GA-based adaptation method are also very small but still visible. In other words, two different adaptation methods results in slightly different engine models with slightly different adaptation accuracies. In the ICM-based method the largest adaptation error for the individual target performance parameters is below.% and the RMS for all the target performance parameters is around 5.79e. In the GA-based adaptation method the maximum error for the individual target performance parameters is around.% and RMS for all the target performance parameters is around.. (9), doi:.6/j.apenergy.9.5.

8 8 Y.G. Li, P. Pilidis / Applied Energy xxx (9) xxx xxx Initial and Adaptation Error (%) Initial ICM-based GA-based - T6 P6 T P Ts mf T P Fig.. Adaptation accuracies of two adaptation methods using Set A. Adaptation Error (%) T6 P6 Set A Set D T P Fig.. Comparison of adaptation accuracies using Set A and Set D in GA-based adaptation. Ts mf T P... Robustness The comparison of the two adaptation methods shows that the ICM-based adaptation method normally provides more accurate adaptation results but the convergence is very sensitive to the selection of the to-be-adapted component parameters []; i.e. if the most sensitive component parameters are included in the tobe-adapted parameter set the ICM-based method will converge, otherwise it may diverge. On the contrary, the GA-based adaptation method provides slightly less accurate adaptation results but is much more robust compared to the ICM-based method; i.e. it can easily provide a converged result for different selection of the to-be-adapted component parameters with appropriate upper and lower bounds. To demonstrate the above features, two sets of the to-beadapted component parameters (Sets A and D in Table ) are selected where Set A includes all the potential to-be-adapted component parameters while Set D includes all except TET and g t. When the ICM-based adaptation method is used, Set A provides an excellent solution shown in Figs. and but Set D does not even provide a converged result due to that two most sensitive component parameters (TET and g t ) are excluded. It shows that if the most sensitive to-be-adapted component parameters are not included the ICM-based adaptation method may diverge and therefore no result will be obtained. When the GA-based adaptation method is applied, the same two sets of the to-be-adapted component parameters all provide results although the result from Set D is much poorer than that from Set A. The comparison of the adaptation accuracies from Set A and D are shown in Fig. where the result from Set D has bigger adaptation errors compared to that from Set A. This is because the quality of the solutions is strongly affected by the selected to-beadapted component parameters and their corresponding upper and lower bounds that determine the adaptation searching space. However it shows that the GA-based adaptation method is numerically much more robust than the ICM-based adaptation method.... Computing speed The ICM-based adaptation method is very fast in computation. With a typical desktop PC computer using Pentium IV processor, it takes about s per iteration and around 7 s for the whole calculation with 6 iterations in total. A typical convergence process for the ICM-based adaptation calculation is shown in Fig.. Comparatively, the GA-based adaptation method is much more time consuming than the ICM-based adaptation method. A typical GA searching process for an adaptation calculation is shown in Fig. for a total number of GA generations. By using the same computer, it takes about s for one GA generation and 5 min RMS and s for a complete adaptation calculation with the final calculation setting shown in Table.. Conclusions Iterations Fig.. Convergence process of ICM-based adaptation: RMS vs. iterations (total of 7 s). Fitness Generations Fig.. GA-based adaptation searching process: GA fitness vs. GA generations (total of 5 min and s). In this paper, a novel GA-based non-linear design-point performance adaptation method has been introduced and described in (9), doi:.6/j.apenergy.9.5.

9 Y.G. Li, P. Pilidis / Applied Energy xxx (9) xxx xxx 9 details and is compared with the ICM-based adaptation method. Application and comparison of the two adaptation approaches to the real performance status estimation of an industrial gas turbine engine show that: Design-point performance adaptation can be regarded as an optimization problem and Genetic Algorithm is one of the best optimizers to serve such purpose. It has been proved to be a very effective and robust method in the performance adaptation. There is no restriction in the selection of the to-be-adapted component parameters for the GA-based adaptation while the searching space and therefore the quality of the adaptation are determined by the selected to-be-adapted parameters and their corresponding upper and lower bounds. With a good selection of the to-be-adapted component parameters and proper upper and lower bounds, the GA-based adaptation method is able to provide satisfactory adaptation solutions. The determination of the upper and lower bounds of the to-be-adapted component parameters can only be made based on trial and error to ensure that the searching space defined by the upper and lower bounds cover the final results of the to-be-adapted component parameters. Application of the GA-based adaptation method to an industrial gas turbine application shows that the GA Fitness can achieve as high as.99 and the maximum adaptation error for individual target measurable parameters can be as low as.5%. The calculation time for the GA-based adaptation is relatively long due to the nature of GA algorithms compared to other numerical searching or optimization methods. However, the computation time required is perfectly acceptable in engineering applications. The stochastic nature of GA algorithms only causes slightly different adaptation solutions if the same GA-based adaptation calculations are repeated, showing that the GA-based adaptation method is very reliable in providing good adaptation results. Comparing with the ICM-based adaptation method, the GA-based adaptation method is numerically more robust in providing an adapted performance model than the ICM-based adaptation method although the prediction accuracy is slightly lower. The ICM-based adaptation method is much faster in computation than the GA-based adaptation method; for the design-point adaptation application in the test case demonstrated in this study the ICM-based method takes about 7 s while the GA-based adaptation method takes about 5 min and s in a final adaptation calculation. Acknowledgements The authors would like to thank MANX Electricity Authority, Isle of Man, UK for allowing us to use their test data and providing valuable technical information. References [] Roth B, Doel D, Mavris D, Beeson D. High-accuracy matching of engine performance models to test data. ASME GT-878, ASME TURBO EXPO, June. [] Roth BA, Doel DL, Cissell JJ. Probabilistic matching of turbofan engine performance models to test data. ASME GT5-68, ASME TURBO EXPO 5, June 5. [] Li YG, Pilidis P, Newby MA. An adaptation approach for gas turbine design-point performance simulation. ASME GT5-68 [or ASME J Eng Gas Turb Power 6;8(October):789 95]. [] Li YG. An genetic algorithm approach to estimate performance status of gas turbines. ASME GT8-575, ASME Turbo Expo 8, June 8. [5] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Addison Wesley Publishing Company Inc.; 989. [6] Jeffs E. Isle of man plant has novel design. Int Turbomach ;5(): 8. [7] Leaflet GE. GE marine & industrial engines: LM5+ gas turbine [or < [8] Macmillan WL. Development of a module type computer program for the calculation of gas turbine off-design performance. Ph.D. Thesis, Cranfield University, England; 97. [9] Li YG, Singh R. An advanced gas turbine gas path diagnostic system PYTHIA. In: ISABE-5-8, ISABE, Munich, Germany, September 5. (9), doi:.6/j.apenergy.9.5.