Wiebe Function Research Articles

Experimental investigation of the effect of altitude on efficiency and emissions of a diesel engine

The Diesel engine is expected to be available for operation at high altitude. However, power loss and emission deterioration have been plaguing highland Diesel engines. Therefore, this study aimed to investigate the impact of altitude on the performance and combustion characteristics of Diesel engines that is limited discussed in existing studies. The research was conducted by varying the altitude from 0-4500 m using a research Diesel engine and analyzing the combustion characteristics at different combustion phases with the help of triple Wiebe function. The results indicated a noticeable drop in power output with increasing altitude, and the deterioration of performance and emissions became significant when the altitude exceeded 3000 m. Specifically, the indicated specific CO, unburned hydrocarbon, and soot emissions increased while nitrogen NOx showed a reverse trend. Additionally, it was found that the lower cylinder pressure at high altitude extended the ignition delay and caused a higher heat release rate in the premixed combustion stage. Moreover, the high altitude condition shortened the duration of combustion and reduced the energy release fraction in the diffusion phase. Furthermore, the late combustion phase occurred earlier and lasted longer at high altitude, which consequently reduced the combustion and thermal efficiency. The most important finding is that the engine performance, especially the combustion efficiency, shows an abrupt degradation with altitudes above 3000 m. As a result, engines operating at extremely high altitudes require multi-stage turbocharging to compensate for combustion deterioration.

Construction and verification of dual-fuel engine combustion model

In order to achieve real-time mapping and online optimization of the combustion process of a dual-fuel engine that is in prolonged operation, this paper is the first to combine the Wiebe function with a deep learning neural network to propose a zero-dimensional (0-D) combustion prediction model for a biodiesel-diesel dual-fuel engine. First, the parameters of the double Wiebe functions are calculated by the Pelican Optimization Algorithm (POA), and the operating parameters and Wiebe parameters are used as input and output parameters of neural networks, respectively, to construct parameter identification models. Then, the combustion process is simplified and reconstructed by combining the Wiebe function with the deep learning neural network, and a 0-D prediction model based on the hybrid model-driven and data-driven method is established, which can further obtain combustion results such as cylinder pressure curve and indicated mean effective pressure (IMEP). The results show that the coefficient of determination (R2) value of the dual-fuel engine 0-D prediction model based on the double Wiebe function combined with POA–CNN–Bi-LSTM is 0.9827, and the model has good prediction accuracy and generalization. The development of the combustion model provides reliable numerical model support for the online evaluation and optimization of dual-fuel engine performance.

Development of a Thermodynamic Model for Supercharged Diesel Engine Performance and Combustion Characteristics Study

ABSTRACT Development of a model that allows performance and combustion characteristics for a supercharged diesel engine was the main objective sought by the present work. Thus, the developed model is used, to examine the impact of start combustion, combustion duration, compression ratio, and heat flux, as well as intake conditions such as pressure and temperature, on the combustion characteristics of the supercharged diesel. For this purpose, a one zone thermodynamic prediction model was adopted with Wiebe function for combustion sub-model. The heat transfer was correlated using Woschni correlation. A numerical simulation is developed considering the crankshaft angle as the independent variable. Validation of the computational code has been favorably evaluated using our experimental data. To give a more general aspect to the developed model, experimental data found in the literature, are also used for this purpose. The results show that the addition of a turbocharger increases low-speed airflow and hence fuel consumption. In addition, an increase in intake pressure contributes to the rise of the heat flux released during combustion, while an increase in intake temperature leads to a strong increase in combustion temperature. Moreover, an increase in the compression ratio leads in a remarkable increase in all parameters simultaneously. However, maximum combustion pressure limits must not be exceeded. This is because the pressure has an effect with the engine mechanical strength.

Estimation of the shape factor (m) in Wiebe function at different operating conditions for a SI engine

The burning process is one of the most important periods, which affects thermal efficiency and exhaust gas emissions, in internal combustion engines. The combustion process in internal combustion engines is modeled with one-dimensional or multi-dimensional software because it is cheaper, faster, and more practical than experiment. One of these methods, which is used to model the combustion period, is the Wiebe function. The Wiebe equation is an approach used in calculating the mass fraction burned and the heat release rate. The selection of Wiebe parameters is one of the most important factors affecting the accuracy of the mass fraction burned. In this study, the measured cylinder pressure of a spark ignition engine was directly used to calculate the heat released rate. The experiments were conducted at different brake mean effective pressures, engine speeds and relative air/fuel ratios, which were called independent variables. The shape factor (m) was determined by fitting the Wiebe equation to the heat release rate curves, which were extracted from the experimental results. The relationship between determined shape factor and independent variables was analyzed with a statistical approach. Eventually, a linear regression model, which explains 80% of the change in the shape factor, was created.

Estimation of Wiebe Function Parameters for Syngas and Anode Off-Gas Combustion in Spark-Ignition Engines

Abstract Wiebe functions, analytical equations that estimate the fuel mass fraction burned (MFB) during combustion, have been effective at describing spark-ignition (SI) engine combustion using gasoline fuels. This study explores if the same methodology can be extended for SI combustion with syngas, a gaseous fuel mixture composed of H2, CO, and CO2, and anode-off gas; the latter is an exhaust gas mixture emitted from the anode of a Solid Oxide Fuel Cell, containing H2, CO, H2O, and CO2. For this study, anode off-gas is treated as a syngas fuel diluted with CO2 and vaporized water. Combustion experiments were run on a single-cylinder, research engine using syngas and anode-off gas as fuels. One single Wiebe function and three double Wiebe functions were fitted and compared with the MFB profile calculated from the experimental data. It was determined that the SI combustion process of both the syngas and the anode-off gas could be estimated using a governing Wiebe function. While the detailed double Wiebe function had the highest accuracy, a reduced double Wiebe function is capable of achieving comparable accuracy. On the other hand, a single Wiebe function is not able to fully capture the combustion process of a SI engine using syngas and anode off-gas.

Quasi-Realistic Performance Analysis of Modern Atkinson Cycle

In this study, a quasi-realistic thermodynamic analysis was performed to investigate the effects of design and operating parameters on the performance of a single-cylinder modern Atkinson cycle engine. Fortran was used for all calculations. It was assumed that the fuel-air mixture was used as the working fluid, and iso-octane was used as the fuel. The Wiebe function was used for the combustion process and it was assumed that the specific heat of the working fluid varies with temperature. In the calculations, heat transfer loss, combustion efficiency, mechanical friction, and pumping losses were taken into account. In the analysis, the closing of the intake valve, equivalence ratio, geometric compression ratio, and the initial conditions of the intake proses were used as independent variables. The effects of these variables on brake mean effective pressure, effective power, specific fuel consumption, and thermal efficiency were investigated. Increasing the inlet pressure, increasing the geometric compression ratio, and delaying the closing of the intake valve increased the mean effective pressure, thermal efficiency, engine output power, and torque. The increase in the inlet temperature adversely affected the engine performance and the specific fuel consumption increased. Engine performance parameters worsened when the equivalence ratio fell below 0.8 and rose above 0.9.

Calibration of 0-D combustion model applied to dual-fuel engine

To acquire a highly accurate 0-D combustion model of the Wiebe function, a genetic algorithm (GA) based on algebraic analysis is developed. By analyzing the in-cylinder combustion process of biodiesel and diesel, a new method is proposed for determining the number of Wiebe functions and the transition angle between premixed and diffusive combustion in dual-fuel engine. First, for different operating conditions, the number of required Wiebe functions and the initial value of Wiebe parameters are determined by algebraic analysis method. Then, genetic algorithm is applied to obtain the further optimized Wiebe parameters and finally compared with Levenberg-Marquardt (LM) algorithm on fitting precision. The algorithm is applied to a dual-fuel engine under the conditions of propeller performance. The results show that, based on the initial value of Wiebe parameters, the fitting process of both LM algorithm and genetic algorithm converges rapidly, the R2 of xb are all at a high level (larger than 0.997), but the RMSE values of genetic algorithm are all in a low range (smaller than 0.013). The fitting effect of genetic algorithm is obviously better than that of LM algorithm. Therefore, genetic algorithm based on algebraic analysis is an incredibly precise algorithm for structuring a 0-D combustion model.

A New Simple Function for Combustion and Cyclic Variation Modeling in Supercharged Spark Ignition Engines

Research in the field of Internal Combustion (IC) engines focuses on the drastic reduction of both pollutant and greenhouse gas emissions. A promising alternative to gasoline and diesel fuel is represented by the use of gaseous fuels, above all green hydrogen but also Natural Gas (NG). In previous works, the authors investigated the performance, efficiency, and emissions of a supercharged Spark Ignition (SI) engine fueled with mixtures of gasoline and natural gas; a detailed research involving the combustion process of this kind of fuel mixture has been previously performed and a lot of experimental data have been collected. Combustion modeling is a fundamental tool in the design and optimization process of an IC engine. A simple way to simulate the combustion evolution is to implement a mathematical function that reproduces the mass fraction burned (MFB) profile; the most used for this purpose is the Wiebe function. In a previous work, the authors proposed an innovative mathematical model, the Hill function, that allowed a better interpolation of experimental MFB profiles when compared to the Wiebe function. In the research work presented here, both the traditional Wiebe and the innovative Hill function have been calibrated using experimental MFB profiles obtained from a supercharged SI engine fueled with mixtures of gasoline and natural gas in different proportions; the two calibrated functions have been implemented in a zero-dimensional (0-D) SI engine model and compared in terms of both Indicated Mean Effective Pressure (IMEP) and cyclic pressure variation prediction reliability. It was found that the Hill function allows a better IMEP prediction for all the operating conditions tested (several engine speeds, supercharging pressures, and fuel mixtures), with a maximum prediction error of 2.7% compared to 4.3% of the Wiebe function. A further analysis was also performed regarding the cyclic pressure variation that affects all the IC engines during combustion and may lead to irregular engine operation; in this case, the Hill function proved to better predict the cyclic pressure variation with respect to the Wiebe function.

Investigation of the Effect of Simulated Atmospheric Conditions at Different Altitudes on the Combustion Process in a Heavy-Duty Diesel Engine Based on Zero-Dimensional Modeling

Abstract Plateau diesel engines suffer from increased fuel consumption and deteriorating emissions. These problems were mainly caused by the combustion worsening inside the chamber. However, limited research has been done specifically on the effect of altitude on the combustion process of diesel engine cycles, which would obviate the feasibility of optimizing high altitude engines. As a result, the goal of this study was to apply the zero-dimensional modeling approach to deeply analyze the influence of altitude on diesel engine combustion. A triple Wiebe function model was calibrated based on the experimental results of a turbocharged direct injection compression ignition engine operating at simulated atmospheric conditions from sea level to 5000 meters, where only the effect of pressure reduction was considered, but the intake temperature and oxygen to nitrogen ratio were kept constant. The analyses indicated that the increase in altitude lengthened the ignition delay, resulting in more fuel fraction being burned in the premixed combustion stage and therefore extending the duration of this phase. As for the main mixing-controlled combustion phase, operation at high altitude retarded the combustion initiation angle, advanced the combustion end angle, shortened the burning duration, and reduced the diesel mass burned in this stage. Moreover, the higher altitude operation increased the energy release and prolonged the duration of the late combustion period, which was detrimental to clean emissions. All these impacts contributed to the reduced thermal and combustion efficiency of the highland engines. However, the engine phasing did not change with increasing altitude, suggesting that it was mainly the combustion degradation that caused the reduction in power output. Consequently, finding solutions to improve the spray formation quality or the spray spatial distribution in lower density backgrounds is the key to compensate for the altitude negative effects.

A Double-Wiebe Function for Reactivity Controlled Compression Ignition Combustion Using Reformate Diesel

AbstractReactivity controlled compression ignition (RCCI) combustion has previously been proposed as a method to achieve high fuel conversion efficiency and reduce engine emissions. A single-fuel RCCI combustion strategy can have decreased fuel system complexity by using a reformate fuel for port fuel injection and the parent fuel (diesel) for direct injection. This paper presents a one-dimensional computational model of a compression ignition engine with single-fuel RCCI. A Wiebe function is used to predict the combustion process by representing the mass fraction burned (MFB) on a crank angle resolved basis. One single-Wiebe function (SWF) and two double-Wiebe functions(DWFs) were fitted to experimentally derive MFB data using the least-square method. The fitted results were compared with MFBs calculated from experimental data to verify the accuracy. The SWF did not fully capture the MFB curve with high fidelity while the detailed DWF captured the MFB curve within a root mean square error of 1.4%. The reduced double-Wiebe function (RDWF) also resulted in a predicted combustion profile with similar accuracy. Hence, the RDWF was used in a GT-power thermodynamic study to understand the effects of the low-temperature heat release (LTHR) fraction and combustion phasing on combustion characteristics. At optimum phasing of 5–10 crank angle degree after the top dead center, increasing the LTHR fraction from 20% to 60% resulted in the fuel conversion efficiency increasing from 39.5% to 41.1%, thus suggesting that the reformate fuel-based RCCI strategy is viable to unlock improved combustion performance.

The Engine Combustion Phasing Prediction Based on the Support Vector Regression Method

While traditional one-dimensional and three-dimensional numerical simulation techniques require a lot of tests and time, emerging Machine Learning (ML) methods can use fewer data to obtain more information to assist in engine development. Combustion phasing is an important parameter of the spark-ignition (SI) engine, which determines the emission and power performance of the engine. In the engine calibration process, it is necessary to determine the maximum brake torque timing (MBT) for different operating conditions to obtain the best engine dynamics performance. Additionally, the determination of the combustion phasing enables the Wiebe function to predict the combustion process. Existing studies have unacceptable errors in the prediction of combustion phasing parameters. This study aimed to find a solution to reduce prediction errors, which will help to improve the calibration accuracy of the engine. In this paper, we used Support Vector Regression (SVR) to reconstruct the mapping relationship between engine inputs and responses, with the hyperparametric optimization method Gray Wolf Optimization (GWO) algorithm. We chose the engine speed, load, and spark timing as engine inputs. Combustion phasing parameters were selected as engine responses. After machine learning training, we found that the prediction accuracy of the SVR model was high, and the R2 of CA10−ST, CA50, CA90, and DOC were all close to 1. The RMSE of these indicators were close to 0. Consequently, SVR can be applied to the prediction of combustion phasing in SI gasoline engines and can provide some reference for combustion phasing control.

Study on control-oriented emission predictions of PPCI diesel engine with two-stage fuel injection

Model predictive control (MPC) is an effective technology used to improve the control accuracy of low temperature combustion engines. The main purpose of this study is to establish a control-oriented emission prediction model based on fuel injection parameters, which provides the basis for the development of MPC controller for PPCI engines. In this study, the combustion model and backpropagation (BP) neural network emission prediction model of a partially premixed compression ignition (PPCI) diesel engine were established, through which the real-time combustion and emissions can be realized by inputting two-stage fuel injection parameters. The Wiebe function linearization method was proposed for the combustion model to achieve high-precision reconstruction of combustion at different fuel injection parameters. The nitrogen oxide (NOx) and particulate matter (PM) emissions were predicted using the combustion characteristics calculated by the combustion model. The results show that the established combustion model can accurately reconstruct the heat release rate and cylinder pressure of PPCI combustion, and the maximum error between the calculated and test values of the combustion parameters is less than 9%. The three-layer BP neural network can achieve high-precision predictions of NOx and PM emissions, with a prediction accuracy greater than 95%. Under the 300th continuous transient seconds, the cumulative prediction error of NOx was 2%, and the cumulative prediction error of PM was 4.9%, indicating high accuracy. This provides the basis for developing MPC controllers for PPCI engines.

Dynamic modeling of a free-piston engine based on combustion parameters prediction

A dynamic combustion model (DCM) is proposed for model-based control synthesis design in an opposed-piston free-piston engine (FPE). The DCM combines a modified triple Wiebe function with an artificial neural network (ANN), and considers the effects of engine variables such as cylinder wall temperature, fuel injection quantity and ignition timing. Firstly, it was proved by algebraic analysis that the triple Wiebe function can well describe the three combustion stages of the FPE. The ANN was then trained using the data obtained through CFD simulations at different operating points, with the aim of predicting combustion parameters based on engine variables. After the model was calibrated, it was verified by the test results from a prototype. Furthermore, this model was used to investigate combustion control strategies for cold start and stable operation. The results showed that the DCM has high precision and can simulate the complex combustion process under a wide range of FPE conditions, and is a favorable tool to make control strategies without experiments, especially in the cold start phase. Unlike general combustion modeling methods, this study provides a cost-effective way to build such a model for controller design.

Knock recognition based on vibration signal and Wiebe function in a heavy-duty spark ignited engine fueled with Methane

Increasing demands on higher performance and lower fuel consumption and emissions have lead the path for internal combustion engine development; this race is nowadays directly related of CO2 emissions reduction. In spark-ignited (SI) engines, knock is one of the major barriers to achieve high thermal efficiency at high loads. The knocking risk is even higher in heavy-duty (HD) engines due to the size of the cylinders and to the low rotation speed. This paper proposes a knock detection strategy based on the combination of knock sensors and combustion modeling applied to a HD natural gas (NG) engine. The aim is to have a reliable, economic and computationally efficient algorithm to be implemented directly on the engine ECU.The method proposed has been applied to an extensive set of experimental data acquired on a SI NG heavy-duty engine. The results of the proposed knock estimation method are benchmarks with those based on in-cylinder pressure analysis using piezoelectric transducers. The extension of the method based on in-cylinder pressure to a high displacement heavy-duty NG engine not only represents an innovation, but improves the knock recognition based on in-cylinder pressure compared with conventional methods as MAPO or IMAP. Besides, the development of an alternative method based on knock sensor signal, allows to obtain a higher or equal sensitivity compared to the traditional MAPO method based on in-cylinder pressure, with the advantage of only using knock sensors.

A novel approach for experimental study and numerical modeling of combustion characteristics of a hydrogen fuelled spark ignition engine

This work is aimed for experimental study on effects of spark timings, compression ratios (4.5:1 to 7.2:1) and exhaust gas recirculation (EGR) on combustion characteristics of hydrogen fuelled spark ignition (SI) engine operating at 3000 rpm. The results indicate spark timing should be reduced for hydrogen fuelled engine as compared to the gasoline fuelled engine. Both ignition lag and DOC decreased with an increase in equivalence ratio and compression ratio, while the opposite trend was observed with an increase in EGR. The EGR was limited up to 23.5 % by volume, as higher EGR lead to cyclic variation. The standard Wiebe function constants were modified for the calculation of mass fraction burnt in a hydrogen fuelled SI engine. In addition to this, the empirical correlations were developed for ignition lag (IL) and duration of combustion (DOC) as a function of compression ratio, equivalence ratio, and EGR. The notable findings that emerged from this study are that hydrogen-fuelled SI engines should be operated at varying spark timing with respect to varying equivalence ratios (0.5 to 0.8) and EGR up to 23.5% by volume.

A Simulation of Four Stroke Marine Diesel Engines to Predict Internal Cylinder Characteristics

This paper introduces a simulation of four-stroke marine diesel engines. The submodel of a particular cylinder was carried out, based on the first law of thermodynamics, programmed by Matlab/Simulink program, which describes the relations among internal characteristics, including cylinder performance parameters, heat release, heat loss, and pressure. The heat release is based on the Wiebe function and the heat loss is based on the Woschni function to build submodels. From the result of the model, the indicated pressure of a single cylinder was taken, the brake power of the engine could be estimated through this pressure. The object of the simulation is a new engine, hence the technical documents and test records provided by the manufacturer are sufficient. The model got the input parameters from this and the key outputs of the model (for example the brake power, peak combustion pressure, specific fuel consumption) were compared with the test records to adjust and make it more accurate. These gaps were not over 5%, therefore, this model can be used to predict key complicated internal cylinder characteristics, for example, the pressure, temperature, and thermal efficiency of engines.

Performance map of a LPG-diesel dual-fuel engine based on experimental and non-linear least squares determined wiebe function

The achievement of sustainable and climate-friendly economies by means of renewable energies, and the transformation into competitive and industrialized economics are among the goals fixed by the African Union (AU) by the year 2063. The implementation of new technologies allowing the achievement of high-level performances in the field of transport and manufacturing industries through low-polluting energy systems is therefore imperative in achieving these objectives. Dual-fuel engines, which combine the low-polluting properties of natural gas and the performance of diesel, present interesting prospects in this regard. Modern engines need to be designed and monitored using an experimentally generated map, which comes at a certain cost. In this work, we present the development of a dual-fuel engine map based on a new model implemented using the nonlinear least squares method in which the Wiebe coefficients are trained for different speeds using Gaussian models. The model is validated with an error of 2% against experimental data. The generated map with the mass fraction of the primary fuel z, equivalence ratio Φ, and engine's speed as varying parameters shows a decrease in specific consumption as z and speed increase. Thermal efficiency decreases as the equivalence ratio increased. Nitric Oxide (NOx) emissions decrease with the increase of Φ and increase with speed, whereas unburnt hydrocarbon (HC) emissions increase with the equivalence ratio and speed. It is found that for an increase in speed of 0.2%, and in z by 2.1%, the specific consumption decreases by 0.1%. An increase in Φ of 0.1% leads to a decrease of NOx emissions by 0.25%. These results comply with previous experimental works, thus showing the relevance of the use of simulation-based engine maps in dual-fuel engine design.

Modification of the Wiebe function for methane-air and oxy-methane- based spark-ignition engines

Predictive combustion models assess engine performance for given setpoints by determining the rate of heat release that is governed by the fuel mass burned. Often, simple mathematical functions such as the Wiebe function are used in simulations to provide mass fraction burned values at each integrated step. Despite its popularity, a limitation of the Wiebe function is that it generally does not take operating conditions into account. Therefore, there is a need for an adaptive Wiebe function that can be scaled for a given fuel and run-time conditions. Here, this is accomplished for methane-air combustion in a spark-ignition (SI) engine. A linear regression analysis was used to fit the adaptive Wiebe model to experimental data with an average R2 value of 0.965. The scaling process was expanded to include oxy-methane combustion for specialized SI applications, e.g., for stationary or mobile power generation in environments such as Mars. This was accomplished through an analysis of the laminar and turbulent flame speeds that dominate the combustion process and the charge burnup time. The results show that a relative increase in the turbulent flame speeds during oxy-methane combustion shrinks the combustion duration between 10% and 90% mass fraction burned by approximately 87% relative to air. Comparing these results with literature reveals that the scaled Wiebe function follows theory closely and is a reasonable tool for burnt mass and combustion duration predictions.

Internal combustion engine diagnostics using statistically processed Wiebe function

The aim of the article is to present the concept of an indirect diagnostic method using the assessment of the variability of the amount of released heat (mass fraction burn) and the heat release rate. The Wiebe function for the assessment of variability has been used. The Wiebe function parameters from the course of the high-pressure indication in the cylinder of internal combustion engine using linear regression have been calculated. From a sufficiently large number of measured samples, the upper and lower limits of the Wiebe function parameters have been statistically determined. Lower and upper limits characterize variability of the heat release process not only in terms of quantity but also in terms of heat release rate. The assessment of variability is thus more complicated than using one integral indicator, typically the mean value of amount of the released heat. The procedure enabling a more accurate estimation of heat generation beginning has been shown. For the combustion process variability assessment of the engine, statistical test of relative frequencies has been used.

Design, development and testing a hybrid control model for RCCI engine using double Wiebe function and random forest machine learning

Reactivity controlled compression ignition (RCCI) engine technology significantly reduces emissions of NOx and soot while improving thermal efficiency. RCCI engine development requires optimum in-cylinder combustion metrics as desired objective across different operating conditions. Physics-inspired control models are a cost-competitive option to achieve these objectives in engine control systems. Combining these control models with machine learning can offer computational efficiency, higher accuracy and lower number of experiments needed for control system calibration. In this context, the present paper proposed a novel hybrid control model by combining physics-inspired parametrized double Wiebe function (D-W) and Random Forest Machine Learning (RFML) to predict both the average and cyclic variation trends of combustion metrics in an RCCI engine. In this work, firstly, an optimized D-W parameters matrix was formulated that can stochastically reconstruct pressure cycles in agreement with 99.7% of the experimental confidence interval. RFML was then employed to develop correlative learning models between experimental control variables such as premix ratio, engine load, injection timing etc. with corresponding D-W parameters matrix. The hybrid control model was used to predict peak pressure (PP), indicated mean effective pressure (IMEP), crank angle for 50% of fuel mass fraction burnt (θ50) and maximum pressure rise rate (MPRR) across 60 test cases and 36000 cycles. Results showed an error mean and standard deviation (μerr±σerr) of 0.068 ± 1.840 bar, 0.007 ± 0.264 bar, 0.15 ± 0.98° CA and 0.018 ± 0.221 bar/° CA between predictions and experiment for PP, IMEP, θ50 and MPRR, respectively in the tenable RCCI operation regime. Proposed hybrid control model exhibited good accuracy and was able to capture the local cyclic variation trends.