Modelling and Prediction of Diesel Engine Performance using Relevance. Vector Machine

Transcription

1 Modelling and Prediction of Diesel Engine Performance using Relevance Vector Machine Ka In Wong a, *, Pak Kin Wong a, Chun Shun Cheung b a Department of Electromechanical Engineering, University of Macau, Macau b Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong *Corresponding author. Tel: ; fax: address: imkain@gmail.com (Ka In Wong) Abstract Diesel engines are being increasingly adopted by many car manufacturers today, yet no exact mathematical diesel engine model exists due to its highly nonlinear nature. In the current literature, black-box identification has been widely used for diesel engine modelling and many artificial neural network (ANN) based models have been developed. However, ANN has many drawbacks such as multiple local minima, user burden on selection of optimal network structure, large training data size and over-fitting risk. To overcome these drawbacks, this paper proposes to apply an emerging machine learning technique, relevance vector machine (RVM), to model and predict the diesel engine performance. The property of global optimal solution of RVM allows the model to be trained using only a few experimental data sets. In this study, the inputs of the model are engine speed, load and cooling water temperature, while the output parameters are the brake specific fuel consumption and the 1

2 amount of exhaust emissions like nitrogen oxides and carbon dioxide. Experimental results show that the model accuracy is satisfactory even the training data is scarce. Moreover, the model accuracy is compared with that using typical ANN. Evaluation results also show that RVM is superior to typical ANN approach. Keywords: diesel engine modelling, engine performance, artificial neural network, relevance vector machine, data scarcity Nomenclature ANN BPNN BSFC CO 2 NO x RBFNN RMSE RMSE test RMSE train R 2 R 2 test Artificial neural network Back-propagation neural network Brake-specific fuel consumption Carbon dioxide Nitrogen oxides Radial basis function neural network Root mean square error Root mean square error for testing data Root mean square error for training data Coefficient of determination Coefficient of determination for testing data 2

3 R 2 train RPM RVM Coefficient of determination for training data Revolution per minute Relevance vector machine 1. Introduction Air pollution is a serious problem in many cities. Despite the tightening of emission standards, the amount of exhaust emissions is still increasing as more and more motor vehicles are being used on roads. According to a recent study (Lim et al. 2012), more than 3.2 millions of people around the world died prematurely in 2010 because of the ambient air pollution problem, mainly caused by diesel soot and exhaust gases. On the other hand, there is also a demand on the reduction of fuel consumption so as to reduce reliance on fossil fuel. Therefore, many researchers have been working on the improvement of vehicle engines, aiming to reduce the exhaust emission and fuel consumption concurrently. Diesel engines are almost exclusively used on commercial vehicles like buses and trucks due to their high fuel efficiency and high durability. However, they are the major emitter of many pollutant gases, including carbon dioxide (CO 2 ) and nitrogen oxides (NO x ). The former is a global warming gas while the latter is posing significant threat to human health. These two gases together bring along both health and environmental problems. Therefore, in order to reduce these emissions and fuel consumption simultaneously, the combustion process of 3

4 the diesel engine must be well controlled. To do so, more dedicated investigations have to be conducted to understand the performance and emission characteristics of the diesel engine. Obviously these investigations are complicated, expensive and time-consuming (Canakci, Erdil, and Arcaklioǧlu 2006; Wong et al. 2010). A way to solve these problems is to create a mathematical diesel engine model so that all the costly and immeasurable data can be predicted and virtual sensors can be built to replace the costly sensors (Hanzevack et al. 1997). Figure 1 is an example of diesel engine performance map originated from (Heywood 1988), which shows that the relationship among the engine torque, engine speed and brake specific fuel consumption is already highly nonlinear. It could be imagined that if the emissions and combustion parameters are studied together, the model will be even more complicated. For example, the combustion process of a diesel engine includes sub-processes like fuel spray, fuel ignition and flame propagation, which are too complex for building up an exact mathematical model. Although several analytical multi-zone models of diesel engine have been developed (Hiroyasu, Kadota, and Arai 1983; Babajimopoulos et al. 2005; Rakopoulos, Antonopoulos, and Rakopoulos 2007), these models require many assumptions and human knowledge in defining those engine-specific parameters, making themselves constrictive in application. Black-box identification, as its name, can manage complex and uncertain information 4

5 without any knowledge of its internal workings. It is, therefore, very suitable for engine modelling and prediction. Many recent researches in black-box identification have already described the use of artificial neural network (ANN) on the modelling of diesel engine performance based on experimental data. For instance, Traver, Atkinson, and Atkinson (1999) described the use of three different types of ANNs as a virtual sensor for the prediction of diesel engine emissions. Najafi et al. (2007) conducted an analysis of a commercial diesel engine with the aid of an ANN model that was developed based on the experimental data collected from that engine. Canakci, Erdil, and Arcaklioǧlu (2006), Ghobadian et al. (2009) and Aydogan, Altun, and Ozcelik (2011) also constructed biodiesel engine models using ANN with back-propagation algorithm. More recently, Ismail et al. (2012) still used ANN to predict different engine responses for a diesel engine fuelled with biodiesel blends. Many other related works of diesel engine modelling using ANN could still be found in the current literature (Oǧuz, Saritas, and Baydan 2010; Ozgur et al. 2011; Yusaf, Yousif, and Elawad 2011). Although ANN has been adopted in many researches, according to the studies of Haykin (1999) and Wong et al. (2010), ANN has, in general, three main drawbacks for its learning process, which make these ANN models not favorable for practice use. These drawbacks are summarized as follows: 5

6 1) The architecture of ANN, including the number of hidden neurons, has to be determined a priori or modified while training by heuristic, which results in a sub-optimal network structure. 2) The training process (i.e. the minimization of the residual squared error cost function) in ANN can easily become stuck in local minima. Various ways of preventing local minima, like early stopping, weight decay, have been employed. However, these methods greatly affect the generalization of the estimated function (i.e., the capacity of handling new input cases). 3) The amount of training data is usually large for ANN. Normally, ANN require at least 200 to 400 sets of training data to build an accurate gasoline engine model (Wong et al. 2010). However, in the case of diesel engine, the collection of the engine emission and performance data is very time-consuming and costly, and the range of the engine operating speed is narrow too. Therefore, the size of the representative training data set is always very small, resulting in that ANN may not be a good solution for diesel engine modelling. Fortunately, an algorithm entitled relevance vector machine (RVM) was recently proposed by Tipping (2001) to overcome the aforementioned drawbacks of ANN. It is an approach based on sparse Bayesian inference, so the solutions are obtained probabilistically and the 6

7 corresponding weights are highly sparse. The most compelling feature of RVM is that it reduces the effort of tuning numerous parameters as required by traditional ANN. The training algorithm of RVM also ensures a global optimal solution (i.e., good generalization), whereas the learning process of ANN may lead to a local optimal solution, which is the reason why ANN requires more representative training data to minimize the risk (Haykin 1999). With these advantageous properties, RVM do not require too much sample data to build an accurate model. Although one deficiency of this approach is that the training time is approximately in the cube of the sample numbers, an accelerated training algorithm was further developed by Tipping and Faul (2003). In this algorithm, an empty model is initialized first. Then, within the same principal framework, samples are sequentially added to increase the marginal likelihood or deleted to avoid any redundant, while their weights are modified at the same time. Owing to the accelerated algorithm, RVM becomes very favorable for modelling and prediction. It was recently adopted by many researchers for system modelling and predictive control problems. For example, Khalil et al. (2005) utilized RVM and other modelling approaches for the simulation and prediction of contaminant levels in groundwater. Yuan et al. (2007) applied RVM to the modelling of seed-separating process. Wong et al. (2012) also compared RVM with other modelling approaches on the modelling and control of a piezostage. These researches show that RVM is generally superior to the ANN. Nevertheless, 7

8 the application of RVM to the modelling of diesel engines is still very few. For these reasons, RVM is employed to model and predict the diesel engine performance in the present study. Experiments are conducted to collect sample data for RVM model training and validation. An ANN-based diesel engine model is also constructed based on the same sample data set and compared with the RVM model in order to demonstrate the effectiveness of this approach. 2. Experimental setup for sample data collection RVM, like other black-box identification method, is data-driven, so experiments were conducted to collect sampling data for model training and validation. A naturally aspirated, water-cooled, 4-cylinder, direct-injection diesel engine was employed for the experiment. The engine specification is shown in Table 1, and the experimental setup is illustrated in Figure 2. In the setup, the engine was connected to an eddy-current dynamometer with a control system used for adjusting its speed and torque. Ultra low sulfur diesel fuel containing less than 10-ppm-wt sulfur was used for data sampling. Anapol EU5000 exhaust gas analyzer was used to measure the gaseous species in the engine exhaust on a continuous basis. It used infra-sensors for measuring CO 2 concentrations and used chemical cells for measuring nitrogen monoxide (NO) and nitrogen dioxide (NO 2 ) to obtain the NO x concentration. The gas analyzer was calibrated with standard and zero gases before each experiment. The resolution of the equipment is summarized in Table 2. 8

9 The experiments were conducted at engine speeds of 1200, 1400, 1600, 1800, and 2000 rpm, and each at engine torque of 28, 70, 140, 210, and 252 Nm. At each speed and torque, data were recorded after the engine had reached the steady state, which was indicated by the lubricating oil temperature and the cooling water temperature. For the purpose of reducing experimental uncertainties, the data were recorded continuously for 5 minutes and each test was carried out three times and the average values were used. Besides, in each test, the volumetric flow rate of fuel was measured using a measuring cylinder and then converted into mass consumption rate based on the density of the fuel (i.e., fuel flow rate). The brake specific fuel consumption (BSFC) was then derived based on the engine speed, engine torque and fuel flow rate. From the experiments, only 22 sets of data corresponding to different engine speeds and torque were collected. From the view point of the machine learning approaches, 22 sets are considered as scarce. This problem of scarce data set is always encountered in small-scale test laboratories. The collected sampling data, including the BSFC, CO 2 and NO x emissions, corresponding to different load and speed settings, are summarized in Table Diesel engine modelling 3.1. Model parameters Before constructing the diesel engine model, the input and output parameters must be 9

10 defined in advance. Since 22 data sets were collected from the experiments, to separate them, 18 of them were used as the training data for the model construction, and the rest of 4 sets were used as testing data for model evaluation. The two controllable parameters, engine speed and engine load, are selected as the input parameters. The cooling water governs the engine temperature, so the cooling water temperature is considered as an important factor and thus it is treated as the input parameter in this study too. For the output parameters, the measured data like BSFC, CO 2 and NO x emissions are chosen. engine speed i.e., = engine torque coolant temperature BSFC, = CO 2 NO x (1) 3.2. Modelling using RVM According to the theory of RVM (Tipping 2001; Tipping and Faul 2003), the diesel engine model can be approximated by Eq. (2): = +, = (2) where is the prediction output of the RVM model for the unseen input data, is the kth input vector of the training data sets, =,, is the weight vector of the RVM model,, is the kernel function and = 1,,,,,. Gaussian radial basis function was selected as the kernel function because it can easily fit scattered and highly nonlinear behavior and usually outperforms other mapping in regression problem 10

11 domain (Seeger 2004)., = (3) In order to construct an accurate model, the weight vector in Eq. (2) and the basis width ( ) of the kernel function need to be well estimated. In the RVM algorithm, this can be done by firstly determining the hyperparameter vector =,, through the maximization of the likelihood function as formulated by Eq. (4). = 1 2 Nlog 2 +log + +y + y (4) where is defined as diag,,, and y is the output vector of the training data sets. The maximization of Eq. (4) over is known as the type-ii maximum likelihood procedure, which can be accelerated using the strategy in (Tipping and Faul 2003), where an empty model is first initialized and then samples within the same principal framework are sequentially added or deleted to increase the marginal likelihood. After the most probable is generated from the procedure, it is then put into Eq. (5) to evaluate the posterior mean : = + y. (5) The posterior mean calculated from Eq. (5), which consists of very few non-zero elements (i.e., highly sparse), is used as the estimation of the weight vector. As a result, the RVM diesel engine model can be further defined as: 11

12 = +. (6) From the above RVM algorithm, only the value of the basis width need to be defined by the user. In this study, this value is tuned by using leave-one-out cross-validation, which is a well-known validation scheme specifically suitable for scarce dataset. Before training the RVM model, each input and output value, say, in the data sets was normalized according to the procedure in (Pyle 1999) in order to increase the model accuracy and prevent any parameter from dominating the output values. The normalization range is [0, +1], which can be done using Eq. (7). = = (7) where, and are the normalized parameter, the upper limit of the input/output parameter before normalization and the lower limit of the input/output parameter before normalization, respectively. After the model is trained, the output values predicted by the model need to be de-normalized using the inverse of Eq. (7) to obtain the actual values. The above RVM modelling algorithm, including the validation and the normalization scheme, was implemented using MATLAB R2012a and executed under Windows 7 on a computer with Intel Core i7 processor and 6GB RAM onboard to build the diesel engine model. 12

13 3.3. Model evaluation To illustrate the performance of the RVM model, the model predicted output values are compared with the actual values from the experimental data sets (i.e., desired values). Two performance indices, namely root mean square error ( ) and the coefficient of determination ( ), are used to evaluate the model, which can be calculated using Eqs. (8) and (9). = 1 (8) =1, where = 1 (9) In Eqs. (8) and (9), is the th model predicted value, is the desired value corresponding to the inputs that give, is the mean of the desired values, and is the number of data points. It has to be noticed that, a smaller means a better model accuracy, whereas a higher means the better the model performs in prediction. Moreover, for comparison purpose, another diesel engine model was constructed using a multilayer feed-forward ANN with back-propagation based on the same sampling data sets. Since multilayer feed-forward neural network is a well-known universal estimator (Bishop 1995) and many researches for diesel engine modelling (Traver, Atkinson, and Atkinson 1999; Khalil et al. 2005; Canakci, Erdil, and Arcaklioǧlu 2006; Najafi et al. 2007; Ghobadian et al. 13

14 2009; Oǧuz, Saritas, and Baydan 2010; Yusaf, Yousif, and Elawad 2011; Aydogan, Altun, and Ozcelik 2011; Ozgur et al. 2011; Ismail et al. 2012) were done based on this configuration, the results from it can be considered as a rather standard benchmark. The structure of the ANN used is listed in Table 4, which is very similar to those ANN structure proposed by other researchers. The RMSE and R 2 for the prediction results of both the RVM model and ANN model are summarized in Table 5. To make the modelling results more readable, Figures 3 to 5 depict the comparison between the models predicted values and the desire values. By comparing the modelling results in the figures, one can learn that RVM is, obviously, superior to ANN, and that the RVM diesel engine model constructed is reliable Discussion of the modelling results From Table 5, it can be seen that RVM outperforms the ANN by 92.13% in RMSE train, 54.02% in RMSE test, 16.41% in R 2 train, and 31.40% in R 2 test. The relatively high RMSE train of the ANN shows that the data sets are not sufficient for building such a highly-nonlinear model. This agrees with the previous studies that ANN usually requires a large sample data size to train an accurate model. Moreover, RVM attempts to optimize global parameters, whereas ANN easily stuck in local minima. Hence, the prediction result of ANN is worse than RVM. Furthermore, only one parameter,, needs to be tuned by the user for RVM, 14

15 while the learning rate, number of hidden layers and number of hidden neurons are required in ANN, which means a grid of guessed values for these parameters have to be prepared and examined. The RMSEs of both RVM and ANN for predicting the BSFC are relatively large as compared to the other output parameters. This is because the function of BSFC is extremely complicated, only 18 training data sets are insufficient for both modelling algorithms, but it is believed that the model accuracies can be improved by increasing the number of training data. Overall, the prediction accuracy of RVM for scarce data is satisfactory. 4. Conclusions In this study, RVM has been applied to model the diesel engine performance and emission characteristics under the condition of data scarcity. Although the combustion process of the diesel engine is unknown, the RVM model has successfully demonstrated the relation between the input parameters, namely the engine speeds, engine loads and cooling water temperature, and the output variables, including the BSFC, CO 2 and NO x concentrations. Experimental results show that the RVM model is still satisfactory even the available data sets are very few. It is believed that more training data sets can improve the model accuracy. Furthermore, a comparison between the RVM model and an ANN model has also been conducted. The results indicate that the average accuracy of the RVM model is higher than 15

16 that of the ANN model, implying that RVM is superior to the ANN. With the diesel engine model constructed by RVM, experimental efforts can be reduced significantly as the performance and emissions of the diesel engine can be predicted more easily. A more comprehensive diesel engine model will be further developed if other engine performance data, such as hydrocarbon emissions, particulate mass concentration, in-cylinder pressure and heat release rate, are available. By applying the RVM model as a virtual sensor on diesel vehicles, the exhaust emissions can be controlled more effectively by incorporating with some advanced control algorithms, such as model predictive control. The study of model predictive diesel emission control based on RVM model will be considered as a future work. Since RVM can also perform online model update, the applications of RVM to online system modelling and online control will also be explored in the future. Acknowledgements The research is supported by the University of Macau Research Grant, grant numbers MYRG149(Y2-L2)-FST11-WPK, and the short-term visiting scholar programme of University of Macau. The authors would like to thank the support from The Hong Kong Polytechnic University and the technician, Mr. Hang Cheong Wong, of the Automotive Engineering Laboratory of University of Macau. 16

17 References Aydogan, H., A. A. Altun, and A. E. Ozcelik "Performance analysis of a turbocharged diesel engine using biodiesel with back propagation artificial neural network." Energy Education Science and Technology Part A: Energy Science and Research no. 28 (1): Babajimopoulos, A., D. N. Assanis, D. L. Flowers, S. M. Aceves, and R. P. Hessel "A fully coupled computational fluid dynamics and multi-zone model with detailed chemical kinetics for the simulation of premixed charge compression ignition engines." International Journal of Engine Research no. 6 (5): Bishop, C.M Neural networks for pattern recognition. New York: Oxford University Press. Canakci, M., A. Erdil, and E. Arcaklioǧlu "Performance and exhaust emissions of a biodiesel engine." Applied Energy no. 83 (6): Ghobadian, B., H. Rahimi, A. M. Nikbakht, G. Najafi, and T. F. Yusaf "Diesel engine performance and exhaust emission analysis using waste cooking biodiesel fuel with an artificial neural network." Renewable Energy no. 34 (4): Hanzevack, Emil L., Theresa W. Long, Chris M. Atkinson, and Michael L. Traver "Virtual sensors for spark ignition engines using neural networks." Proceedings of the 1997 American Control Conference no. 1:

18 Haykin, S Neural Networks: A comprehensive foundation. New Jersey: Prentice-Hall. Heywood, John B Internal Combustion Engine Fundamentals. New York: McGraw-Hill. Hiroyasu, Hiroyuki, Toshikazu Kadota, and Masataka Arai "Development and use of a spray combustion modeling to predict diesel engine efficiency and pollutant emissions (part 1 combustion modeling)." Bulletin of the JSME no. 26 (214): Ismail, H. Mohamed, H. K. Ng, C. W. Queck, and S. Gan "Artificial neural networks modelling of engine-out responses for a light-duty diesel engine fuelled with biodiesel blends." Applied Energy no. 92: Khalil, A., M. N. Almasri, M. McKee, and J. J. Kaluarachchi "Applicability of statistical learning algorithms in groundwater quality modeling." Water Resources Research no. 41 (5). Lim, S. S., T. Vos, A. D. Flaxman, et al "A comparative risk assessment of burden of disease and injury attributable to 67 risk factors and risk factor clusters in 21 regions, : A systematic analysis for the Global Burden of Disease Study 2010." The Lancet no. 380 (9859): Najafi, G., B. Ghobadian, T. F. Yusaf, and H. Rahimi "Combustion analysis of a CI engine performance using waste cooking biodiesel fuel with an artificial neural network 18

19 aid." American Journal of Applied Sciences no. 4 (10): Oǧuz, H., I. Saritas, and H. E. Baydan "Prediction of diesel engine performance using biofuels with artificial neural network." Expert Systems with Applications no. 37 (9): Ozgur, T., G. Tuccar, M. Ozcanli, and K. Aydin "Prediction of emissions of a diesel engine fueled with soybean biodiesel using artificial neural networks." Energy Education Science and Technology Part A: Energy Science and Research no. 27 (2): Pyle, D Data preparation for data mining. San Francisco: Morgan Kaufmann. Rakopoulos, C. D., K. A. Antonopoulos, and D. C. Rakopoulos "Development and application of multi-zone model for combustion and pollutants formation in direct injection diesel engine running with vegetable oil or its bio-diesel." Energy Conversion and Management no. 48 (7): Seeger, M "Gaussian processes for machine learning." International Journal of Neural Systems no. 14 (2): Tipping, M. E "Sparse Bayesian Learning and the Relevance Vector Machine." Journal of Machine Learning Research no. 1 (3): Tipping, M.E., and A.C. Faul Fast marginal likelihood maximisation for sparse Bayesian models. Paper read at Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, at Key West, FL. 19

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21 List of figures Figure 1. Performance map for an air-cooled naturally aspirated diesel engine (Heywood 1988) Figure 2. Schematic diagram of experimental setup Figure 3. Comparison between the models predicted values and the desired values of BSFC Figure 4. Comparison between the models predicted values and the desired values of CO 2 Figure 5. Comparison between the models predicted values and the desired values of NO x 21

22 Figure 1. Performance map for an air-cooled naturally aspirated diesel engine (Heywood 1988) 22

23 Dynamometer control system Eddy-current dynamometer Data acquisition system Diesel engine Exhaust gases Gas analyzers Figure 2. Schematic diagram of experimental setup 23

24 Predicted value of BSFC (g/kwh) RVM ANN Desired value of BSFC (g/kwh) Figure 3. Comparison between the models predicted values and the desired values of BSFC 24

25 Predicted value of CO 2 (%) RVM ANN Desired value of CO 2 (%) Figure 4. Comparison between the models predicted values and the desired values of CO 2 25

26 Predicted value of NO x (ppm) RVM ANN Desired value of NO x (ppm) Figure 5. Comparison between the models predicted values and the desired values of NO x 26

27 List of tables Table 1. Engine specifications Table 2. Resolution of the equipment Table 3. Experimental data for model training and validation Table 4. Architecture of the ANN model Table 5. Prediction results of the RVM model and the ANN model 27

28 Table 1. Engine specifications Model Type Maximum power Maximum torque Bore stroke Displacement Isuzu 4HF1 In-line four-cylinder 88 kw / 3200 rpm 285 Nm / 1800 rpm 112 mm 110 mm 4334 cc Compression ratio 19.0:1 Fuel Injection timing (BTDC) 8 Injection pump type Injection nozzle Bosch in-line type Hole type (with five orifices) 28

29 Table 2. Resolution of the equipment Equipment Resolution Speed control 1 rpm Ono Sokki diesel engine system Torque control 0.1 Nm Throttle control 0.1% CO 2 analyzer 0.01% NO x analyzer 1 ppm Measuring cylinder 1% 29

30 Table 3. Experimental data for model training and validation Data set number Engine speed (rpm) Engine torque (Nm) Coolant temperature ( o C) BSFC (g/kwh) CO 2 concentration (%) NO x concentration (ppm)

31 Number of hidden layer 1 Number of hidden neurons 20 Table 4. Architecture of the ANN model Activation function for hidden layer Activation function for output layer Adaption learning function Hyperbolic tangent sigmoid transfer function (tansig) Linear transfer function (purelin) Gradient descent with momentum weight/bias learning function (LEARNGDM) Training function Levenberg-Marquardt method (TRAINLM) 31

32 Table 5. Prediction results of the RVM model and the ANN model Output parameters Model RMSE train * RMSE test * R 2 train R 2 test BSFC CO 2 concentration NO x concentration Overall Average RVM ANN RVM ANN RVM ANN RVM ANN * Smaller is better 32