tial component parameters against the engine performance parameters. The adaptation .... evolution and is one of the optimization searching techniques. A.
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
GA-based design-point performance adaptation and its comparison with ICM-based approach Y.G. Li *, P. Pilidis School of Engineering, Cranfield University, Bedford MK43 0AL, United Kingdom
a r t i c l e
i n f o
Article history: Received 18 February 2009 Received in revised form 17 April 2009 Accepted 19 May 2009 Available online xxxx Keywords: Gas turbine Engine Performance adaptation Performance matching Design-point performance simulation Influence Coefficient Matrix (ICM) Genetic Algorithm (GA)
a b s t r a c t 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 2009 Elsevier Ltd. All rights reserved.
1. Introduction 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 par-
* Corresponding author. Tel.: +44 1234 754723; fax: +44 1235 751566. E-mail address: i.y.li@cranfield.ac.uk (Y.G. Li).
ties 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 [1] for engine cycle model matching and a minimum variance estimator algorithm [2] for performance matching of a turbofan engine. Li et al. introduced an Influence Coefficient Matrix
0306-2619/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.05.034
Please cite this article in press as: Li YG, Pilidis P. GA-based design-point performance adaptation and its comparison with ICM-based approach. Appl Energy (2009), doi:10.1016/j.apenergy.2009.05.034
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Nomenclature Symbols a ACM GA H h( ) ICM M ma mc1/mc2 N OF P PR SFC T TET Ts ~ x ~ x^ ~ z
~ z^ weighting factor Adaptation Coefficient Matrix Genetic Algorithm Influence Coefficient Matrix vector-valued function Influence Coefficient Matrix number of target performance parameters engine inlet air flow rate (kg/s) cooling flow rate to turbines 1 and 2 number of to-be-adapted component parameters Objective Function total pressure (atm) pressure ratio specific fuel consumption (mg/N s) total temperature (K) Turbine entry temperature (K) static temperature (K) actual component parameter vector estimated to-be-adapted component parameter vector actual target performance parameter vector
(ICM) based adaptation method [3] 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 [4]. 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 [3] is carried out and is applied to the design-point performance status estimation of an industrial gas turbine engine GE LM2500+ running at Manx Electricity Authority (MEA), UK. The advantages and disadvantages of both adaptation methods are discussed. 2. 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 hð~ xÞ
ð1Þ
e D D P Pb
r g
simulated target performance parameter vector adaptation error deviation Burner relative pressure loss (%) summation convergence threshold component isentropic efficiency
Subscripts b Burner c1 Compressor 1 c2 Compressor 2 t1 turbine 1/compressor turbine t2 turbine 2/power turbine 0 baseline point 6 Compressor 2 exit 11 compressor turbine exit 13 nozzle exit 20 cooling flow from Compressor 1 exit
where ~ z 2 RM is the target performance parameter vector and M the number of the target performance parameters, ~ x 2 RN 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. 2.1. 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. 1. At certain ambient and operating condition and for an engine with ^ simulated engine certain design-point component parameters ~ x; 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 x^ whose initial value should be given to component parameters ~ 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
Simulated performance zˆ
Target performance
z GA
Fitness=Max?
Engine Model No Estimated to-be-adapted
Yes
parameters
xˆ
Stop Fig. 1. GA-based design-point adaptation method.
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OF ¼
M X ai z^i zi 100 M zi i¼1
ð2Þ
where zi are the actual values of the target performance parameters and z^i are the corresponding simulated performance parameters from the engine performance model. ai 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 ai = 1.0 (i = 1–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:
ei ¼
z^i zi 100% zi
ð3Þ
where ei is the adaptation error for ith 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
1 1 þ OF
ð4Þ
where the Objective Function (OF) defined in Eq. (2) represents the difference between predicted and actual measurements. Thus, a value of Fitness approaching 1 indicates a good solution while a value of Fitness approaching 0 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 1 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
Table 1 GA parameter setting. No.
Parameters
1 2 3 4
Population size No. of generations Probability of cross-over Probability of mutation
Value Initial calculation
Final calculation
20 20 0.4 0.3
100 100 0.6 0.7
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]. 2.2. 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. 2.3. ICM-based adaptation method An Influence Coefficient Matrix (ICM) based design-point performance method was published by Li et al. [3] for the same perfor-
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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. (1) 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:
z ¼ H D~ D~ x
ð5Þ
^ D~ ^ H is Influence Coefficient Matrix where D~ z ¼~ z ~ z; x ¼~ x ~ x; (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 per^ may be different from their actual values formance parameters (~ z) ~ (z) and such a difference is the prediction error of the performance z. Therefore, parameters relative to their target represented by D~ 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) H1 leading to Eq. (6) when N = M:
x ¼ H1 D~ z D~
ð6Þ
If N > M, Eq. (5) is under-determined and a pseudo-inverse is defined as:
H# ¼ HT ðHHT Þ1
ð7Þ
If N < M, Eq. (5) is over-determined and the pseudo-inverse is defined as: #
T
1
H ¼ ðH HÞ H
T
ð8Þ
The solution resulting from the pseudo-inverse
z D~ x ¼ H# D~
RMS ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uP 2 u M z^i zi t i¼1 z 100 i
M
