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MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 2020, VOL. 26, NO. 2, 119–143 https://doi.org/10.1080/13873954.2020.1713821 ARTICLE Modelling the clogging of gas turbine ﬁlter houses in heavy- duty power generation systems a a Sabah Ahmed Abdul-Wahab , Abubaker Sayed Mohamed Omer , b b Kaan Yetilmezsoy and Majid Bahramian Department of Mechanical and Industrial Engineering, College of Engineering, Sultan Qaboos University, Muscat, Sultanate of Oman; Department of Environmental Engineering, Faculty of Civil Engineering, Yildiz Technical University, Istanbul, Turkey ABSTRACT ARTICLE HISTORY Received 18 June 2019 A prognostic approach based on a MISO (multiple inputs and single Accepted 7 January 2020 output) fuzzy logic model was introduced to estimate the pressure diﬀerence across a gas turbine (GT) ﬁlter house in a heavy-duty KEYWORDS power generation system. For modelling and simulation of clog- Clogging phenomenon; ging of the GT ﬁlter house, nine real-time process variables (ambi- fuzzy logic model; heavy- ent temperature, humidity, ambient pressure, GT produced load, duty power generation inlet guide vane position, airﬂow rate, wind speed, wind direction system and PM10 dust concentration) were fuzziﬁed using a graphical user interface within the framework of an artiﬁcial intelligence-based methodology. The results revealed that the proposed fuzzy logic model produced very small deviations and showed a superior pre- dictive performance than the conventional multiple regression methodology, with a very high determination coeﬃcient of 0.974. A complicated dynamic process, such as clogging phenomenonin heavy-duty GT system, was successfully modelled due to high cap- ability of the fuzzy logic-based prognostic approach in capturing the nonlinear interactions. 1. Introduction In industry, the demand for internal combustion engines continues to rise signiﬁcantly for generating electricity and operating machinery. Gas turbines (GT) (Figure 1) are internal combustion engines that use ambient air as a working ﬂuid. In these systems, air is compressed by the fan blades as it enters the engine from the intake part, resulting in an increase in the pressure through the compressor. The compressed air is mixed and burned with fuel in the combustion chamber to release energy. The hot exhaust gases provide forward thrust and turn the turbines which drive the compressor fan blades. This released energy is then utilized to rotate the main shaft, and subsequently, produce energy in the required form [1]. Gas turbines are commonly paired with steam turbines through a combined cycle arrangement in power plants where the exhaust heat of the gas turbine is used to power the steam turbine to achieve desired production eﬃciency. For instance, Mitsubishi Heavy Industries plant initially produced 1050 MW of electrical CONTACT Kaan Yetilmezsoy yetilmez@yildiz.edu.tr; kyetilmezsoy@gmail.com Department of Environmental Engineering, Faculty of Civil Engineering, Yildiz Technical University, Davutpasa, Esenler, Istanbul 34220, Turkey © 2020 Informa UK Limited, trading as Taylor & Francis Group 120 S. A. ABDUL-WAHAB ET AL. Figure 1. Stages of a typical gas turbine engine. energy using six steam turbines (single cycle) with an eﬃciency of 43%, and currently uses only three combined cycle arrangements and produces up to 1500 MW of energy with an eﬃciency of 59.1% [2]. GT power plants, unlike steam turbine power plants, do not require much water, because a condenser is not needed in their conﬁguration; hence, they oﬀer an economic advantage [3]. Another advantage of the GT power plants is its ability to accept most commercial fuels to be mixed with air to produce energy and its capability to be switched on and oﬀ in a matter of minutes, making them quite favoured in power plants. The GT air intake system is mainly composed of six elements, namely; ﬁlter house, silencers, elbow, vertical duct, intake dampers and intake duct. The air is drawn, and its quality is ensured by passing it through the GT ﬁlter house. The air then reaches the silencers, which are composed of a single row of silencing baﬄes ﬁtted inside a silencer casing to insulate the noise. The elbow is bolted to the casing of the silencer to guide and direct the air stream to ﬁt the vertical duct dimensions. The vertical duct guides the air coming from the elbow to ﬁt the dimensions of the compressor air intake ﬂap. In the vertical duct, the air meets the intake dampers, which have no function while the turbine is operational. They are only used when the turbine is oﬀ to isolate the area after the vertical duct to prevent ventilation and moisture from entering the compressor. Then, the air passes the intake duct, which is ﬁtted with an inlet cone and attached to the ﬂange of the compressor and has two sections upper and lower. The upper section’s function is to provide the combustion air to the compressor with no turbulence, while the lower section is ﬁxed to the foundation of the air intake system and its function is to provide a connection to the air dryer and detergent for the cleaning process of the compressor blades. The air ﬁnally leaves the intake air system when reaching the inlet cone of the compressor where it will encounter the inlet guide vane (IGV) of the compressor and subsequently, complete its path in the GT system. The function of the air intake system is fully dependent on the ﬁlter house performance because once the ﬁlter house is clogged, the ﬁlter modules must be replaced, and they must ﬁt the air intake system exactly [4,5]. The resistance of ﬁlter media against airﬂow results in a diﬀerence in static pressure between the input and output faces of the ﬁlter, which is called as pressure drop. Pressure drop is a key parameter in the choice of ﬁlters and can cause irreversible mechanical resistance problems beyond a certain value. This parameter has a crucial impact on the MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 121 performance of a ﬁlter medium, and a rapid increase in the pressure drop is associated with the partial or total clogging of the ﬁlter medium [6]. As a result, the clogging of a GT ﬁlter house will obstruct the operation of the system due to the reduction in ﬁltration area caused by the accumulation of dust particles on the surface of the ﬁlter. The clogging will also cause a decrease in the air mass ﬂow downstream the ﬁlter house, which will ultimately diminish the produced load of the GT system [7]. Some investigations were conducted on the eﬀect of the pressure drop in the GT system performance [8,9]. Moreover, past literature investigated the eﬀects of some ambient parameters on ﬁlter clogging, such as humidity [10–14] and temperature [14]. Nevertheless, the existing literature is still short of documented studies researching the eﬀects of several ambient and GT system parameters simultaneously in this particular area. The GT ﬁlter house is essential to ensure a suﬃcient quality of combustible air without damaging the components of the GT system. For instance, corrosion and fouling of GT blades will negatively aﬀect the productivity and eﬃciency of the GT system. A fully clogged ﬁlter house will terminate the power generation operation because clogging will prevent air from reaching the combustion chamber, resulting in no released energy. A clogged ﬁlter house must be replaced immediately to resume the GT operation. Therefore, development of a representative model will be very useful (i) to determine the eﬀects of several ambient conditions and GT system-related parameters on the clogging of the GT ﬁlter house; and (ii) to help pinpoint the reasons that cause high- pressure diﬀerences across the ﬁlter house. Apart from the above-mentioned studies [8,9,12–14], several successful modelling attempts on the ﬁltration/clogging process can be found in the literature [10,15–21]. Furthermore, Sreenuch et al. [22] proposed a scalable data-driven degradation pattern (a parameterized Takagi–Sugeno fuzzy model) to highlight the potential of the prognostic approach in a real-world ﬁlter-clogging case study. In real-time condition monitoring, the authors used a ﬁlter-clogging experiment as an aerospace application, and the degradation and model parameter were simulta- neously forecasted online according to noisy measurement updates using a particle ﬁlter. However, to the best of the authors’ knowledge, there are no systematic papers speciﬁ- cally devoted to the development and implementation of a more practical artiﬁcial intelligence-based model (within the framework of a Mamdani-type fuzzy logic metho- dology) for prediction of the pressure diﬀerence across a heavy-duty GT ﬁlter. Therefore, without requiring a complex model structure, tedious parameter estimation procedures (e.g. random-walk and kernel smoothing parameter estimations) and detailed ﬁrst- principles calculations, the present approach can be considered as a unique attempt for modelling of the clogging of an industrial-scale GT ﬁlter house system. It is worthy to mention that a deep learning has never been applied to ﬁlter-clogging problematic. Nevertheless, the proposed modelling strategy is applicable for a speciﬁc installation with a proper learning process (for strictly maintenance purpose, not for designing the ﬁltration strategy). In consideration of the foregoing facts, the speciﬁc objectives of this study were as follows: (1) to estimate the pressure diﬀerence across a heavy-duty GT ﬁlter house by means of a fuzzy logic-based model consisting of several important real-time process variables (i.e. ambient temperature, pressure and humidity, dust concentration, wind speed and direction, compressor airﬂow rate, inlet guide vane (IGV) position and GT produced load) which aﬀect the clogging of a GT ﬁlter house; and (2) to compare the 122 S. A. ABDUL-WAHAB ET AL. proposed artiﬁcial intelligence-based methodology against the conventional multiple regression-based method for various descriptive statistical indicators, such as coeﬃcient of determination (R ), mean absolute error (MAE), root mean square error (RMSE), systematic and unsystematic RMSE (RMSE and RMSE , respectively), index of agree- S U ment (IA) and fractional variance (FV). 2. Materials and methods 2.1. Gas turbine ﬁlter house and operating conditions The studied heavy-duty gas turbine system is a product of Siemens company model SGT5-4000F with a frequency of 50 Hertz (Hz) and a production capacity of 329 MW (Figure 2). The GT works at a speed of 3000 revolutions per minute and a pressure ratio of 20.1:1. The GT system is equipped with an air intake system produced by BIS Gas Turbine Systems Company GmbH, which supplies the compressor of the GT with a maximum air pressure drop of 10 mbar. The GT ﬁlter house consists of weather hoods, bird screens and ﬁlter modules (Figure 2). The weather hoods and bird screens are installed to protect the intake air from falling rain and to prevent birds from nesting at the ambient intake area, respectively. The Siemens turbine used in the present study has been located in Oman. The ﬁlter modules consist of a self-cleaning pulse ﬁlter system, air preheater, evapora- tive cooler and ﬁne ﬁltration stage. The pulse ﬁlter system (type: CKD 351-900-140-34, dimension: Ø355 × 900 mm, pressure drop (new ﬁlter): 150 Pa, pressure drop (soiled ﬁlter): >1000 Pa, elements: 1176 pieces, supplier: Nordic Air Filtration) removes dust particles by passing the air through the ﬁlter cartridges that have a ﬁlter medium covering their outer surfaces, where more than 96% of the dust particles are removed from the air stream. It is noted that the governing mechanisms of capturing coarse particles are interception, inertial impaction, diﬀusion and sieving [23]. On the other hand, sieving is rarely the dominant mechanism for a ﬁbrous ﬁlter. Since the studied ﬁlter system is not only a deep ﬁlter, so the cake formed on the medium should have an important contribution to the ﬁlter clogging. As time passes the accumulation of the dust particles on the ﬁlter cartridge media may damage the ﬁlter, so that the studied GT system must have its pulse ﬁlters replaced on a yearly basis at ΔP of 900 kPa. The pulse ﬁlter of the SGT5-4000F intake system costs about 74,000 USD. The pressure drop of the pulse ﬁlter system is 1.5 mbar for a clean ﬁlter and more than 10 mbar for a soiled ﬁlter. After a preset time, a pulse cleaning cycle starts to clean the ﬁlter cartridges with nozzles that blow compressed air from the opposite side of the air stream to push out the particles that are stuck on the ﬁlter media when the pressure sensors read a set value of pressure drop. After the pulse cleaning cycle, an automatic conveying cycle starts to collect the dust particles into a bin, and this is done by the dust evacuation system. The particles that are captured in the pulse ﬁlter stage are PM particles, which are fractions of airborne particulate matter with a nominal mean aero- dynamic diameter equals to 2.5–10 µm. The air preheater is only worked when the turbine is operating at low-temperature degrees to prevent icing on the ﬁne ﬁlter stage, and it consists of a heat exchanger operated by warm water to heat the air stream. MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 123 Airflow Direction Figure 2. Flow chart of the studied heavy-duty gas turbine system. In the evaporative cooler, the clean air is cooled through adiabatic humidiﬁcation which will cause a rise in the air density and ﬂow rate to achieve higher GT production. In the downstream of the evaporative cooler, the system is equipped with a droplet-catcher to remove moisture from the air stream. The ﬁne ﬁlter (where the removal of particles larger than 2 µm occur) has a dust removal eﬃciency of 93%, and the pressure drop of the air stream (when the ﬁlter is clean) ranges from 0.9 to 1.1 mbar but reaches a value of 6 mbar after it is soiled. The maximum allowable value of the total dust concentration in the air downstream the ﬁlter house is 0.008 ppm for the particles with a diameter less than 10 µm. The ﬁlter house is only operational under these conditions: (i) the pressure drop across the air intake system is lower than 13 mbar; (ii) the pressure drop across the pulse ﬁlter system is less than 9 mbar; (iii) the pressure drop across the ﬁne ﬁlter stage is less than 4.5 mbar; (iv) the air preheating system is non-operational and the air intake duct is open; (v) the ambient wind speed does not exceed 87.5 knots; and (vi) the air dew point, wet bulb 124 S. A. ABDUL-WAHAB ET AL. and dry bulb temperatures are within the following ranges, respectively: −40–80°C, 0– 100°C and −50–60°C. 2.2. Data collection and descriptive statistics Two groups of data (ambient and GT system data) were collected to conduct the computa- tional analysis of the gas turbine ﬁlter clogging. All ambient data were recorded on a daily basis from the Directorate General of Meteorology of Oman. The dust concentration data were provided by the Ministry of Environment and Climate Aﬀair. The GT system data were collected from the plant control oﬃce on a daily average basis. The pressure diﬀerence across the ﬁlter house is measured via delta pressure gauges situated upstream of the pulse ﬁlter stage and downstream of the ﬁne ﬁlter stage. The pressure transmitting devices are equipped with 4 to 20 mA interfaces which are connected to the plant control oﬃce. The compressor IGV position is governed by an actuator and ranges from 0 (fully closed) to 100% (fully open). The airﬂow rate is measured via sensors that are positioned downstream of the IGV at the compressor inlet. For the GT system, the mass ﬂow rates were calculated between 0 and 588.66 kg/h with an average value of 235.53 (± 114.51) kg/h during the measurement period. Descriptive statistics of model variables considered in the present modelling study are summarized in Table 1. Additionally, nonlinear variations of the model components are depicted in Figure 3. 2.3. A brief representation of input and output variables To develop an eﬃcient model to signify the clogging of a GT ﬁlter house, an adequate selection of parameters involved in the modelling is necessary. A thorough process in determining the input and output parameters of the ﬁlter-clogging model was conducted to facilitate the process of recognizing the eﬀect of each parameter on the clogging phenomenon. In this study, nine input parameters were chosen, namely; ambient temperature, pressure and humidity, dust concentration, wind speed and direction, compressor airﬂow rate, inlet guide vane (IGV) position and GT produced load. The pressure diﬀerence across the ﬁlter house was chosen to be the output parameter of the model because the clogging phenomenon is directly proportional to the pressure drop across the ﬁlter house. The present model components are brieﬂy discussed below. Table 1. Data statistics of model variables considered in the present modelling study. #of X Input variables Unit Minimum Maximum Mean SD X Ambient temperature °C 20.44 37.55 28.90 4.48 X Humidity % 17.54 85.96 62.33 13.39 X Ambient pressure mbar 992.01 1018.90 1006.20 7.44 X Gas turbine produced load MW 0.35 236.32 172.79 49.08 X Inlet guide vane (IGV) position % 0.00 76.33 42.07 16.91 X Air ﬂow rate m /s 0.00 605.57 482.91 128.30 X Wind speed knots 2.85 11.77 5.35 1.22 X Wind direction angle 30.00 360.00 132.89 111.80 X PM dust concentration µg/m 0.00 335.59 135.87 51.21 9 10 #of Y Output variable Unit Minimum Maximum Mean SD Y Pressure diﬀerence across ﬁlter house mbar 0.14 4.71 3.61 1.01 Standard deviation. MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 125 40 100 () b () a () c 0 20 40 60 80 100 120 140 160 180 200 220 240 0 20 40 60 80 100 120 140 160 180 200 220 240 0 20 40 60 80 100 120 140 160 180 200 220 240 Observation number Observation number Observation number X : Ambient temperature (°C) X : Humidity (%) X : Ambient pressure (mbar) 1 2 3 250 80 () e 40 300 50 100 () f () d 0 0 0 20 40 60 80 100 120 140 160 180 200 220 240 0 20 40 60 80 100 120 140 160 180 200 220 240 0 20 40 60 80 100 120 140 160 180 200 220 240 Observation number Observation number Observation number X : Gas turbine produced load X : Inlet guide vane (IGV) X : Air flow rate (m /s) 4 5 6 (MW) position (%) 12 400 () g () i () h 6 150 100 100 2 0 0 20 40 60 80 100 120 140 160 180 200 220 240 0 20 40 60 80 100 120 140 160 180 200 220 240 0 20 40 60 80 100 120 140 160 180 200 220 240 Observation number Observation number Observation number X : Wind speed (knots) X : Wind direction (angle) X : PM dust concentration 7 8 9 10 (µg/m ) () j 0 20 40 60 80 100 120 140 160 180 200 220 240 Observation number Y: Pressure difference across filter house (mbar) Figure 3. Variations of the model components. 2.3.1. Ambient temperature (X ) The ambient temperature is an important parameter because it plays an important role in the air dust in terms of sintering eﬀect which is the compacting of dust particles by forming a mass under high temperatures. Literature also showed that changing the temperature of the air passed through the ﬁlter could directly aﬀect the clogging phenomenon, since it had a direct eﬀect on a ﬁlter mass [14]. Lin et al. [24] proved the association of the ﬁltration eﬃciency with the temperature changes of the passing air. At this point, it is noted that the temperature eﬀect is also associated with thetype ofmaterialtobeusedasthe ﬁlter medium. For example, Lin et al. [24]used Gas turbine produced load (MW) Wind speed (knot) Ambient temperature (°C) Pressure difference across Humidity (%) Wind direction (angle) Inlet guide vane (IGV) position (%) filter house (mbar) PM dust concentration (µg/m³) Ambient pressure (mbar) Air flow rate (m /s) 10 126 S. A. ABDUL-WAHAB ET AL. a tailor-made ceramic ﬁlter element testing system and two ceramic candle ﬁlters (CFs) with diﬀerent ﬁltration characteristics to investigate the eﬀects of various parameters (e.g. temperature, dust concentration and ﬁltration superﬁcial velocity) on the two performance measures (e.g. loading behaviour and dust cakes) of CFs during hot gas ﬁltration process. They reported that the resistance and porosity of dust cakes varied only slightly between 298 K (24.85°C) and 473 K (199.85°C), whereas the resistance abruptly decreased, and the porosity increased when the temperature increased from 473 K to 573 K (299.85°C). In the study, the authors emphasized that the opposite temperature eﬀect on the dust cake structure within a higher temperature range was ascribed to the sintering and dust softening eﬀects. On the other hand, in another study investigating the eﬀect of temperature during cellulose compression, Vaca-Medina et al. [25]reportedadensiﬁcation between 25° C and 60°C for the commercial cellulose powder samples with diﬀerent crystallinity levels. They also observed a diminution of cellulose sorption capacity (particularly for themoreamorphous andpolydispersesamplessuch as α-Cellulose) due to reduction of the speciﬁc surface area after compression at 160°C. According to the pressure- volume-temperature (PVT) diagrams (normalized isobaric curves) presented in their study, the authors concluded that events of this nature could be characteristic of a sintering mechanism. 2.3.2. Humidity (X ) In humid conditions, the ﬁlter media and dust particles tend to absorb the water molecules in the air, which will cause the particles to swell causing a change in their sizes; hence, humidity is a critical factor to consider in GT ﬁlter clogging. Several studies reported a signiﬁcant rise in pressure diﬀerence across a ﬁlter media when exposed to humid air [12,13,26]. 2.3.3. Ambient pressure (X ) The ambient pressure contributes to dust particles coalescence and consequently aﬀects the clogging of the ﬁlter house. It was proven experimentally that varying the air pressure severely aﬀected the ﬁltration eﬃciency of a ﬁlter medium. In a laboratory-scale analysis on the eﬀectiveness of ﬁbrous ﬁlters under diﬀerent air pressures, Xu et al. [27] reported that a noticeable reduction in ﬁltration eﬃciency (up to 15%) and increase in pressure drop were observed, when the air pressure was increased from 600 to 1300 mbar. They have emphasized that the ﬁltration eﬃciency was highly aﬀected by the air pressure change at smaller particle sizes. 2.3.4. Gas turbine produced load (X ) The GT load is correlated to the other studied parameters and it is directly aﬀected by clogging of ﬁlter house in terms of severe drop at advanced stages of clogging due to insuﬃcient air supply to the combustion chamber [8,9,28,29]. This parameter is critical in modelling of the ﬁlter clogging and determining the magnitude of power produced, since it demands certain values of other parameters that have a direct eﬀect on the development of ﬁlter clogging. MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 127 2.3.5. Inlet guide vane (IGV) position (X ) The IGV position parameter inﬂuences the airﬂow rate to the compressor in a non-linear trend. The function of the IGV is to provide an adequate pressure drop through compressor inlet and motivate a whirl motion of air before reaching the compressor impeller [30]. The studied system employs the IGV position to alert the system before the pressure diﬀerence across the ﬁlter house reaches a critical point by sending out an alarm when the pressure diﬀerence exceeds a value of [(0.5 + 0.5 × IGV position) × 11] mbar. 2.3.6. Airﬂow rate (X ) The ﬁlter media cause resistance which increases the pressure drop; consequently, the airﬂow rate will decrease [7]. The airﬂow rate parameter is important to the study because the production of the GT load is a function of the airﬂow rate that correlates to the evolution of clogging. The airﬂow rate was used as an indicator of clogging in a clogging detector invention [31] by using airﬂow rate sensors that indicate when clogging phenomenon occurs. Additionally, the growth of airﬂow can accelerate the clogging syndrome of the ﬁlter house and increase the pressure diﬀerence especially in humid ambient conditions [28]. 2.3.7. Wind speed (X ) and direction (X ) 7 8 The wind speed and direction parameters were proven not to have a direct correlation with the dust load in the atmosphere unless associated with humidity. It was proven that for wind speeds higher than 7.78 knots and at a minimum elevation of 10 m, relative humidity has a speciﬁc trend with the dust load in the atmosphere. Therefore, wind speed and humidity along with the variation of wind direction should all be considered for relatively accurate forecasting of dust load [26]. Dehghanpour et al. [32] proved that wind speed and direction have direct eﬀects on the development of dust storms, which come with a sudden rise in the number of dust particles and accelerate the clogging of the ﬁlter house. 2.3.8. PM dust concentration (X ) 10 9 PM concentration in the atmosphere is the most critical parameter because this particle size poses a signiﬁcant impact on the majority of the particulate matter (2.5–10 µm) that are stuck on the ﬁlter media. Hinds and Kadrichu [33] reported that a unit mass of dust concentration could increase particle capturing on the ﬁlter surface and accelerate the clogging of the ﬁlter media. Sunnu et al. [34] proved that augmentation of dust loading rate increased the pressure drop across the ﬁlter media and caused clogging to happen in a shorter period. 2.3.9. Pressure diﬀerence across ﬁlter house (Y) When studying the eﬀectiveness of a ﬁlter medium, the main parameter to consider is the pressure diﬀerence across the ﬁlter media. Almost all studies on ﬁltration processes refer to the pressure diﬀerence as the main indicator of ﬁlter clogging (e.g. the greater the pressure diﬀerence and the stronger the clogging) [10,15–18]. In the studied system, a pressure diﬀerence of 13 mbar across GT ﬁlter house is a clear indicator of clogging; consequently, all GT operations are seized until the ﬁlter house is replaced. 128 S. A. ABDUL-WAHAB ET AL. 2.4. Fuzzy logic methodology The key idea in fuzzy logic, in fact, is the allowance of partial belongings of any object to diﬀerent subsets of a universal set instead of belonging to a single set completely [35]. There are basically ﬁve parts of the fuzzy inference process [36]: In the ﬁrst step (fuzziﬁcation), crisp numerical inputs and outputs are divided into diﬀerent fuzzy categories associated with linguistic terms (i.e. A, B, C, too-cold, cold, warm, hot, too-hot, large, small, young, old, etc.), where the output is always a fuzziﬁed degree of a speciﬁc membership function within the range from 0 to 1 [37]. In the second step, the fuzzy operator (AND or OR) in the antecedent is performed in the FIS. Two kinds of built-in AND methods (min (minimum) and prod (product)), and two kinds of built-in OR methods (max (maximum) and probor (the probabilistic OR TM method)) can be used in the Fuzzy Logic Toolbox of MATLAB® [38]. In this study, the prod method was applied for the AND operator in the present FIS. In the third step, an implication process from the antecedent to the consequent is performed in the FIS. This procedure is deﬁned as the shaping of the consequent (a fuzzy set) based on the antecedent (a single number). The input for the implication process is a single number given by the antecedent, and the output is a fuzzy set [36]. In this study, the prod method was applied for the implication process in the present FIS. In the fourth step, aggregation process is performed to fuzzy sets to obtain a single fuzzy set that represents the outputs of each fuzzy rule. The sum method was applied for the aggregation process in the present FIS, as similarly conducted in the previous studies of the third author [39,40]. In the ﬁfth step, the defuzziﬁer produces the crisp values corresponding to the ﬁnal fuzzy outputs as a conclusion [41]. In this study, the centre of gravity (COG or centroid) method was implemented as the most commonly used defuzziﬁcation technique for conversion of the resulting fuzzy outputs from the fuzzy inference engine to a number [42]. Considering Fuzzy logic-based prognostic alogrithm for prediction of ΔP across GT ﬁlter house 1: Create a FIS ﬁle (Name = ‘Filter’) and deﬁne type (Type = ‘mamdani’) of [System] 2: Deﬁne number of inputs (NumInputs), outputs (NumOutputs), and rules (NumRules) 3: Deﬁne AndMethod = ‘prod’, OrMethod = ‘max’, ImpMethod = ‘prod’, AggMethod = ‘sum’ 4: Deﬁne DefuzzMethod = ‘centroid’ 5: Create [Input1], . . ., [Input9] and deﬁne Name = ‘Ta’, . . ., Name = ‘DC’ 6: Create range vectors for all variables, Range = [20.44 37.55], . . ., Range = [0 335.59] 7: Set number of membership functions, NumMFs = 8 8: Create level trapezoidal shaped membership functions, MF1 = ‘A’, . . . MF8 = ‘H’:’trapmf’ 9: Create a rule base ([Rules]), set weights (1) and connection (or, and) 10: Create FIS matrix for ‘Filter’,a= readﬁs(‘Filter’) 11: Store the computed data in the Workspace as structure array using data containers (ﬁelds) 12: Read structure array with ﬁelds of name: ‘Filter’, . . ., rule: [1 × 235 struct] 13: Display FIS input-output diagram, plotﬁs(a) 14: Display all of the membership functions for a given variable, plotmf(a,’input’,1) 15: Display membership functions for output variable . . ., plotmf(a,’output’,1) 16: Display FIS rules, showrule(a) or display a speciﬁc FIS rue, e.g. showrule(a,99) 17: Create a GT system-based data set of variables X1, . . . X9: data = [] 18: Load measured data from MAT-ﬁle into workspace: load ‘data’ 19: Perform fuzzy inference calculations: b = evalﬁs(data,a) 20: Calculate statistics R , R, MAE, RMSE, RMSE , RMSE , MSE, IA, FA2, FV, PSE, CV S U 21: Store the computed data in the Workspace as double-precision ﬂoating-point values MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 129 START INTRODUCE INPUTS CRISP Ta Pa GTL Qa WS WD w IGV DC NUMERICAL INPUTS FUZZIFICATION FUZZY F - Pa F - GTL F - IGV F - Qa F - WS F - WD F - Ta F - w F - DC INPUTS RULE 1: IF .... THEN RESULT 1 FUZZY RULE 2: IF .... THEN RESULT 2 INFERENCE DATA BASE DECISION MAKING LOGIC SYSTEM (FIS) (Mamdani) RULE N: IF .... THEN RESULT N RULE BASE AND METHOD F - (Pressure Difference) FUZZY OUTPUT IMPLICATION METHOD AGGREGATION METHOD DEFUZZIFICATION CENTROID METHOD PROD OPERATOR SUM OPERATOR CRISP Pressure NUMERICAL END difference OUTPUT (mbar) Figure 4. A detailed ﬂowchart of the proposed prognostic approach implemented in this study. the above-mentioned steps, a detailed schematic of the proposed fuzzy logic methodology to predict pressure diﬀerence across GT ﬁlter house is depicted in Figure 4. For the present case, the relevant estimations were attained via the ‘Editor’ window (new M-File) in MATLAB® by implementing some speciﬁc built-in functions (e.g. FISMAT = readﬁs(‘ﬁlename’), OUTPUTS = evalﬁs(INPUTS,FIS)) for the predeﬁned FIS ﬁles (created with the extension .ﬁs). Furthermore, the detailed computational steps of the fuzzy logic-based prognostic procedure are presented as pseudo-code below. 2.4.1. Generation membership functions and fuzzy rules In this study, the estimation of the pressure diﬀerence across the GT ﬁlter house was implemented using trapezoidal-shaped membership functions (trapmf)for the fuzzysub- sets of each model variable. For this purpose, the present fuzzy set categories were expressed in the form of letters (i.e. A, B, C, etc.) to build simple fuzzy rules, as similarly 130 S. A. ABDUL-WAHAB ET AL. performed in the previous studies [37,39,40]. Therefore, the model variables had eight-level trapezoidal-shaped membership functions (trapmf-i,where i = 1,2,3,..., 8)namely A, B, C, D, E, F, G and H instead of long linguistic terms such as moderately low, low, moderate, moderately high, high, very high, etc. For example, a GT system-based data set of ‘X : Ambient temperature (T ) = 23.67°C, X :Humidity (w) = 80.09%, X : Ambient pressure a 2 3 (P ) = 1013.20 mbar, X : Gas turbine produced load (GTL) = 172.34 MW, X :Inlet a 4 5 guide vane position (IGV) = 32.76%, X :Air ﬂow rate (Q ) = 495.79 m /s, X :Wind 6 a 7 speed (WS) = 4.28 knots, X : Wind direction (WD) = 120°, X :PM dust concentration 8 9 10 (DC) = 72 µg/m ,and Y: pressure diﬀerence across the GT ﬁlter house (ΔP) = 2.69 mbar’, the fuzzy rule was coded as ‘If (Ta is B) and (w is G) and (Pa is H) and (GTL is F) and (IGV is D) and (Qa is G) and (WS is B) and (WD is C) and (DC is C) then (Dp is E)’. For the proposed fuzzy logic model, the present fuzzy set categories and the collected GT system data, the Mamdani-type fuzzy rules were constructed, and a total of 235 rules were then established in the If-Then format using the Fuzzy Rule Editor of MATLAB® R2018a software (V9.3.0.713579, 64-bit (win64), Academic Licence Number: 40,578,168, TM MathWorks Inc., Natick, MA) operating on a Casper Excalibur (Intel® Core i7- 7700HQ CPU, 2.81 GHz, 16 GB of RAM, 64-bit) PC. 2.4.2. Fuzziﬁcation of model components In the present study, the FIS (Fuzzy Inference System) Editor GUI (graphical user interface) in the Fuzzy Logic Toolbox (within the framework of MATLAB® R2018a software) was used for modelling and simulation of clogging of the GT ﬁlter house. In this computational analysis, nine input variables (T , ω,P , GTL, IGV, Q , WS, WD, DC) a a a and one output variable (ΔP) were fuzziﬁed with eight-level trapezoidal membership functions using a Mamdani-type FIS Editor. Figure 5 illustrates the shapes and ranges of the input variables (X , X , X , ..., X ), and the output variable (Y). The model variables 1 2 3 9 were fuzziﬁed according to the data set collected from GT ﬁlter house (see Section 2.2, Table 1). Table 2 summarizes the number of trapezoidal membership functions (trapmf) and their ranks for each of the input and output variables considered in the present fuzzy logic-based model. 2.5. Multiple regression-based analysis In addition to the proposed fuzzy logic model, a multiple regression–based analysis was also conducted within the scope of the present study. For comparative purpose, the real- time process data obtained from the studied GT system were imported directly from Microsoft® Excel® Oﬃce 365 (Microsoft Inc., Redmond, WA) which was used as an open database connectivity data source and run under Windows 10 system on the same PC platform which was also used in fuzzy logic modelling. The observed data were then appraised by DataFit® (V8.1.69, Oakdale Engineering, PA, US) multiple regression soft- ware package. In the multiple regression–based analysis, the convergence criteria were selected for the –10 following values of the solution preferences: (a) regression tolerance = 1 × 10 ,maximum number of iterations = 250, and (b) diverging nonlinear iteration limit = 10. When perform- ing the nonlinear regression, Richardson’s extrapolation method was conducted to calculate MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 131 Figure 5. Fuzziﬁcation of GT system-based model variables. numerical derivatives for the solution of the models. The multiple regression–based analysis was conducted based on the Levenberg–Marquardt method with double precision. In the computational analysis, the stepwise selection procedure (SSP) was applied as the combination of the forward selection and backward elimination procedures for variable selection process within the framework of DataFit® software. The SSP begins with a forward step (with no variables in the model). After the forward step, the p-values of the variable coeﬃcients are re-examined, and any insigniﬁcant variables are removed in the backward step. This process continues until no variables are either added or removed from the model. The SSP is more generally popular than either the forward or backward procedures. As the models were solved on the DataFit® numeric computing environment, they were automatically sorted by the program based on the goodness-of-ﬁt criteria into a graphical interface on the DataFit® numeric computing environment. Additionally, regression 132 S. A. ABDUL-WAHAB ET AL. Table 2. Number of trapezoidal membership functions (trapmf) and their ranks for each of the input and output variables considered in the present fuzzy sets. Level of trapezoidal membership functions (trapmf) Model variables AB C D E F G H X = T [16.88 17.88 23 24] [23 24 25 26] [25 26 27 28] [27 28 29 30] [29 30 31 32] [31 32 33 34] [33 34 35 36] [35 36 39.1 40.1] 1 a X = ω [11.08 16.08 19 24] [19 24 29 34] [29 34 39 44] [39 44 49 54] [49 54 59 64] [59 64 69 74] [69 74 79 84] [79 84 87.92 92.92] X = P [983.02 984.02 1000 [1000 1001 1002 [1002 1003 1004 [1004 1005 1006 [1006 1007 1008 [1008 1009 1010 [1010 1011 1012 [1012 1013 1024.80 3 a 1001] 1003] 1005] 1007] 1009] 1011] 1013] 1025.80] X = GTL [−18.3 − 0.30 1 19] [1 19 37 55] [37 55 73 91] [73 91 109 127] [109 127 145 163] [145 163 181 199] [181 199 217 235] [217 235 237.64 255.64] X = IGV [−9.64 −4.14 1 6.5] [1 6.5 12 17.5] [12 17.5 23 28.5] [23 28.5 34 39.5] [34 39.5 45 50.5] [45 50.5 56 61.5] [56 61.5 67 72.5] [67 72.5 80.16 85.66] X = Q [−50 −4 4 50] [−50 −4 4 50] [96 142 188 234] [188 234 280 326] [280 326 372 418] [372 418 464 510] [464 510 556 602] [556 602 609.14 655.14] 6 a X = WS [2.1 2.7 3 3.6] [3 3.6 4.2 4.8] [4.2 4.8 5.4 6] [5.4 6 6.6 7.2] [6.6 7.2 7.8 8.4] [7.8 8.4 9 9.6] [9 9.6 10.2 10.8] [10.2 10.8 12.74 13.34] X = WD [−2 22 38 62] [38 62 86 110] [86 110 134 158] [134 158 182 206] [182 206 230 254] [230 254 278 302] [278 302 326 350] [326 350 370 394] X = DC [−30 −5 5 30] [5 30 55 80] [55 80 105 130] [105 130 155 180] [155 180 205 230] [205 230 255 280] [255 280 305 330] [305 330 341.18 366.18] Y = ΔP [−0.52 −0.22 0.5 0.8] [0.5 0.8 1.1 1.4] [1.1 1.4 1.7 2] [1.7 2 2.3 2.6] [2.3 2.6 2.9 3.2] [2.9 3.2 3.5 3.8] [3.5 3.8 4.1 4.4] [4.1 4.4 5.02 5.32] MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 133 variables and descriptive statistics of the residual errors, such as standard error of the estimate (SEE), sum of residuals (SR), residual average (RA), residual sum of squares (RSS), coeﬃcient of multiple determination (R ), correlation coeﬃcient (R), and adjusted coeﬃ- cient of multiple determination (R ) were also determined to evaluate the performance of adj the models. Moreover, t-ratios and p-values were also calculated for the appraisal of the importance of the regression coeﬃcients. An alpha (α) level of 0.05 (or 95% conﬁdence) was used to determine the statistical signiﬁcance of the model components. 2.6. Measuring of the goodness of the estimate In the present analysis, various descriptive statistical indicators such as coeﬃcient of determination (R ), correlation coeﬃcient (R), mean absolute error (MAE), root mean square error (RMSE), systematic and unsystematic RMSE (RMSEs and RMSEu, respec- tively), mean square error (MSE), index of agreement (IA), the factor of two (FA2), fractional variance (FV), proportion of systematic error (PSE) and coeﬃcient of variation (CV) were used as helpful tools to evaluate the prediction performance of the model [43–46]. Some additional statistical analyses were also implemented using StatsDirect (V2.7.2, StatsDirect, Ltd., Altrincham, Cheshire, UK) statistical software package for the veriﬁcation of the obtained results. Prior to applying parametric (unpaired or two-sample and matched- pair t tests) or non-parametric tests (the Mann–Whitney U (or the Wilcoxon rank-sum) test or the Kruskal–Wallis test with the Conover–Inman method), the Shapiro–Wilk W (named after Samuel Sanford Shapiro and Martin Wilk) and the Levene’s (named after Howard Levene) tests were consecutively conducted as preconditions to ensure that the subsets (i.e. estimated data obtained from the studied GT system and the predicted outputs from the derived multiple regression-based formula) had a normal or non-normal distribution, and variances (or standard deviations) of the paired groups were homogeneous or unequal. In the case of the data sets were not normally distributed, a non-parametric test was applied instead of a parametric test. Mann–Whitney U test (named after Henry Berthold Mann and Donald Ramson Whitney) or the Kruskal–Wallis test (named after William Kruskal and W. Allen Wallis) was used to compare the considered subsets. Test results were assessed by two-tailed p-values to reﬂect the statistical signiﬁcance between paired groups [47]. 3. Results and discussion 3.1. Prediction of pressure diﬀerence (ΔP) in the gas turbine ﬁlter house Results of the regression analysis showed that three multiple regression-based models (MRM) were proposed by the DataFit® software for the estimation of pressure diﬀerence in the GT ﬁlter house: (i)a ﬁrst-order polynomial model with constant term (MRM-1); (ii)a ﬁrst-order polynomial function without constant term (MRM-2); and (iii)an exponential model (MRM-3). Results of the multiple regression–based analysis are summarized in Table 3. For the best-ﬁt model (herein the ﬁrst-order polynomial model with constant term: MRM-1); the multiple regression coeﬃcients (a, b, c, . . ., i), constant term (j), and regression variable results including standard error, t-statistics, and corresponding p- values for each variable (X , X , X , ..., X ) are presented in Table 4. The proposed model 1 2 3 9 134 S. A. ABDUL-WAHAB ET AL. Table 3. Summary of the multiple regression-based results. Regression results Residual statistics Calculation MRM-1 MRM-2 MRM-3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ SEE: Standard error of the estimate 0.4246 0.4237 0.4381 ½ ðY Y Þ a p i¼1 np SR: Sum of residuals P -9.30E-13 -1.43E-03 -1.8571 ðY Y Þ a p i¼1 RA: Residual average -3.96E-15 6.06E-06 -7.90E-03 ðY Y Þ a p i¼1 RSS: Residual sum of squares (absolute) P 40.5612 40.5623 43.1766 ðY Y Þ a p i¼1 hi RSS: Residual sum of squares (relative) P 40.5612 40.5623 43.1766 ðY Y Þ a p i¼1 Y : Actual data point; Y : predicted values; n: number of data points or observations; σ: standard deviation of data point; p: a p number of parameters or variables in the regression model. Table 4. Model components and regression variable results for the best-ﬁt model (MRM-1). Multiple regression coef- ﬁcients and constant a b term Input variables SE t-ratio p-value a = 4.60E-02 X : Ambient temperature (°C) 1.55E-02 2.96264 0.00338 b = 6.75E-04 X : Humidity (%) 2.62E-03 0.258129 0.79654 c = −1.93E-03 X : Ambient pressure (mbar) 9.55E-03 −0.20168 0.84035 d = −1.07E-03 X : Gas turbine produced load (MW) 6.26E-03 −0.17153 0.86396 e = −1.33E-02 X : Inlet guide vane (IGV) position (%) 4.66E-03 −2.86084 0.00462 f = 8.32E-03 X : Air ﬂow rate (m /s) 2.06E-03 4.043417 0.00007 g = 2.98E-03 X : Wind speed (knots) 2.45E-02 0.121375 0.90350 h = −2.88E-04 X : Wind direction (angle) 2.71E-04 −1.06434 0.28832 i = 9.52E-04 X :PM dust concentration (µg/m ) 6.48E-04 1.468327 0.14341 9 10 j = 0.803 10.07714 0.079636 0.93660 a b Standard error, p-values <0.05 are the most signiﬁcant. (the best-ﬁt one) deﬁned as a function of nine operating variables [ΔP = f (T , ω,P , GTL, a a IGV, Q , WS, WD, DC)] is given in Eq. (1). 4:60 6:75 1:93 1:07 1:33 ΔP ðmbarÞ¼ ðT Þþ ðωÞ ðP Þ ðGTLÞ ðIGVÞ a a 2 4 3 3 2 10 10 10 10 10 8:32 2:98 2:88 9:52 þ ðQ Þþ ðWSÞ ðWDÞþ ðDCÞþ 0:803 3 3 4 4 10 10 10 10 (1) It is reported that the larger absolute t-ratio indicates the more signiﬁcant parameter in the regression model. Moreover, the variable with the lowest p-value is considered the most signiﬁcant [47]. According to the absolute t-ratios and the corresponding p-values of the model components in Table 4; airﬂow rate, ambient temperature and inlet guide vane (IGV) position have more importance than other variables for the derived poly- nomial model (with constant term) in prediction of pressure diﬀerence across ﬁlter house. Furthermore, scatter plots of pressure diﬀerence across GT ﬁlter house as a function of each of the predictor variables are illustrated in Figure 6, indicating that all variables exhibit certain importance within their speciﬁc ranges, and none of them could be cut without aﬀecting the outcome of the model. It is also noted that process-related MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 135 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 () a () b () c 0 0 0 18 20 22 24 26 28 30 32 34 36 38 40 10 20 30 40 50 60 70 80 90 990 995 1000 1005 1010 1015 1020 1025 Ambient temperature (°C) Humidity (%) Ambient pressure (mbar) X : Ambient temperature (°C) X : Humidity (%) X : Ambient pressure (mbar) 1 2 3 5 5 5 4 4 3 3 3 2 2 1 1 1 () d () f () e 0 0 0 0 50 100 150 200 250 0 20406080 0 100 200 300 400 500 600 700 Gas turbine produced load (MW) Inlet guide vane (IGV) position (%) Air flow rate (m /s) X : Gas turbine produced load (MW) X : Inlet guide vane (IGV) position (%) X : Air flow rate (m /s) 4 5 6 5 5 5 4 4 4 3 3 3 2 2 2 () h 1 1 () i () g 0 0 0 0 100 200 300 2468 10 12 0 100 200 300 400 PM dust concentration (µg/m³) Wind speed (knot) Wind direction (angle) X7: Wind speed (knots) X8: Wind direction (angle) X9: PM10 dust concentration (µg/m ) Figure 6. Scatter plots of pressure diﬀerence across ﬁlter house as a function of each of the predictor variables. aspects of other factors on the clogging phenomenon across ﬁlter house are fully elaborated in the previous studies [8–21]. 3.2. Appraisal of the prediction accuracy To quantify the performance of the proposed models, the computational results were 2 2 assessed with various descriptive statistical measures, such as R , R , R,MAE, RMSE, adj RMSE ,RMSE , PSE, IA, FV, FA2 and CV (also known as relative standard deviation, RSD), S U for measuring the models’ predictive accuracy. Statistical results are summarized in Table 5. Looking at the statistical outputs and deviations of the developed models (Table 5 and Figure 7), it can be concluded that the proposed fuzzy logic model demonstrated a very satisfactory performance on the prediction of pressure diﬀerence across ﬁlter house compared to the multiple regression–based model (R = 0.9735 for FLM and R = 0.8299 for MRM-1). For the present case, the high value of determination coeﬃcient indicated that only 2.65% of the total variations were not explained by the Pressure difference across Pressure difference across Pressure difference across filter house (mbar) filter house (mbar) filter house (mbar) Pressure difference across Pressure difference across Pressure difference across filter house (mbar) filter house (mbar) filter house (mbar) Pressure difference across Pressure difference across Pressure difference across filter house (mbar) filter house (mbar) filter house (mbar) 136 S. A. ABDUL-WAHAB ET AL. Table 5. Descriptive performance indices for the proposed models (n = 235). Developed models Statistical performance indicators Formulations FLM MRM-1 2 2 Determination coeﬃcient (R ) n 0.9735 0.8299 ðO O ÞðP P Þ i m i m i¼1 SSreg R ¼ ¼ n n P P SS þSS res reg 2 2 ðO O Þ ðP P Þ i m i m i¼1 i¼1 SS =df Adjusted coeﬃcient of multiple determination 2 res e 2 n1 0.9724 0.8232 R ¼ 1 ¼ 1ð1 R Þ 2 adj SS =df nk1 tot t (R ) adj Mean absolute error (MAE) 0.1394 0.3416 MAE ¼ jj P O i i i¼1 0:5 Root mean squared error (RMSE) n 0.1660 0.4154 RMSE ¼ ½ P O i i i¼1 hi 0:5 Root mean squared error - systematic (RMSE ) 2 0.0455 0.1703 S P RMSE ¼ ðÞ P O S i i reg i¼1 hi 0:5 Root mean squared error - unsystematic 2 0.1596 0.3789 RMSE ¼ ðÞ P P U i i (RMSE ) reg U n i¼1 ðÞ RMSE Proportion of systematic error (PSE) S 0.0812 0.2021 PSE ¼ ðÞ RMSE 2 3 Index of agreement (IA) 0.9931 0.9528 ðÞ P O i i 6 7 i¼1 IA ¼ 1 4 5 ðÞ jj P O þjj O O i m i m i¼1 qP ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qP ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 Fractional variance (FV) 0.0323 0.0920 ðÞ OiOm ðÞ PiPm 2ðÞ σoσp σ ¼ σ ¼ FV ¼ o p n1 n1 ðÞ σoþσp Factor of two (FA2) 0.9798 0.9227 1 O 0:5 FA2 ¼ 2:0 n P i¼1 Coeﬃcient of variation (CV, %) CV ¼ðÞ RMSE=O 100 4.5991 11.5098 O, P and n indicate the observed, the predicted and the number of data points, respectively. σ is the standard deviation, and the subscripts i and m indicate the data points and mean, respectively. SS : Total sum of squares (proportional to the variance of the data); SS : Sum of squares of residuals (also referred as the tot res residual sum of squares); SS : Regression sum of squares (also referred as the explained sum of squares); df is the reg e degrees of freedom (n – k – 1) of the estimate of the population error variance (where n is the size of the sample, and k is the total number of explanatory variables in the model without including the constant term); df is the degrees of freedom (n – 1) of the estimate of the population variance of the dependent variable. FLM: Fuzzy logic model; MRM-1: Multiple regression model-1 (herein the ﬁrst-order polynomial model with constant term). fuzzy logic model, while 17.01% of the total variations were not described by the conventional regression approach. Additionally, the very high value of the correlation coeﬃcient (R = 0.9867) signiﬁed an excellent correlation between the measured and the fuzzy logic–simulated data. In addition, the value of adjusted determination coeﬃcient (R = 0.9724) was also very high, showing a high signiﬁcance of the fuzzy logic model adj 2 2 [45]. Liu et al. [48] have reported that the R corrects the R value for the sample size adj and the number of terms in the model. If there are many terms in the model and the 2 2 sample size is not very large, the R may be noticeably smaller than the R [45]. In the adj 2 2 present case, the R was found to be very close to the R value, indicating that the adj appropriateness of the sample size used in the modelling study. Alow valueofthe coeﬃcient of variation (CV = 4.5991%) indicated a very high degree of precision for the proposed fuzzy logic model, as suggested by others [45,49]. As seen from Table 5, other descriptive performance indices also revealed that the developed fuzzy logic model produced smaller errors (MAE = 0.1394 mbar for FLM and 0.3416 mbar for MRM-1, RMSE = 0.1660 mbar for FLM and 0.4154 mbar for MRM-1; RMSE = 0.0455 mbar for FLM and 0.1703 mbar for MRM-1; RMSE = 0.1596 S U MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 137 5 1,5 Multiple regression model (MRM-1) () a Fuzzy logic model 1,0 0,5 0,0 -0,5 1 Observed GT system data -1,0 Multiple regression model (MRM-1) Fuzzy logic model () b 0 -1,5 0 20 40 60 80 100 120 140 160 180 200 220 240 0 20 40 60 80 100 120 140 160 180 200 220 240 Observation number Observation number 5 5 () d () c 4 4 3 3 2 2 1 1 0 0 01234 5 01234 5 Pressure difference across Pressure difference across filter house (mbar): Observed filter house (mbar): Observed Figure 7. Visual comparison of fuzzy logic and the best-ﬁt multiple regression models’ outputs in terms of the pressure diﬀerence across GT ﬁlter house: (a) head-to-head agreement, (b) variation of residuals, (c) linear correlation between observed data and MRM-1 outputs and (d) linear correlation between observed data and fuzzy logic outputs. mbar for FLM and 0.3789 mbar for MRM-1, and PSE = 0.0812 mbar for FLM and 0.2021 mbar for MRM-1) or deviations (IA = 0.9931 for FLM and IA = 0.9528 for MRM-1; FV = 0.0323 mbar for FLM and 0.0920 mbar for MRM-1; FA2 = 0.9798 for FLM and 0.9227 for MRM-1) and exhibited a superior prediction performance on forecasting the pressure diﬀerence across GT ﬁlter house compared to the multiple regression–based model. According to parametric/non-parametric statistical analysis, the results of the Shapiro–Wilk W tests showed no evidence of normality for the paired groups of observed data (W = 0.7031, p < 0.0001 < 0.05, n = 235), multiple regression model outputs (W = 0.5221, p < 0.0001 < 0.05, n = 235) and fuzzy logic-based predictions (W = 0.6882, p < 0.0001 < 0.05, n = 235). Thus, the supposition of normality was not corroborated for all groups in favour of the null hypothesis (H :the sample is taken from a normal distribution, p > α = 0.05) of the Shapiro–Wilk W test, since all samples were not taken from a normal distribution for an alpha (α) level of 0.05 (or 95% conﬁdence). For this reason, non-parametric tests (Mann–Whitney U test and Kruskal–Wallis test) were directly performed to compare the considered subsets (observed data and models’ outputs), without applying the Levene’s (W50) test for conﬁrmation of the equality/homogeneity of variances for the paired groups. For the present case, both the Mann–Whitney U test and the Kruskal–Wallis test (with Pressure difference across Pressure difference across filter house (mbar): MRM-1 outputs filter house (mbar) Pressure difference across filter Residual error: Y - Y (mbar) pred obs house (mbar): Fuzzy logic outputs 138 S. A. ABDUL-WAHAB ET AL. the Conover–Inman method) showed that there were no statistically signiﬁcant diﬀerences between the observed data set and the outputs of both multiple regres- sion–based model (two-sided p = p = 0.081) and fuzzy logic model (two-sided MW KW p = p = 0.9913). For both cases, the null hypothesis (H ) was not rejected in MW KW 0 favour of the alternative hypothesis (H )since the p-value was higher than the chosen α level of 0.05 (or 95% conﬁdence). However, when two-sided p-values are scrutinized, it can be concluded that the probabilities of rejecting the null hypothesis of ‘no diﬀerence between the observed data and the model outputs’ (in other words, accepting the alternative hypothesis of ‘there is a signiﬁcant diﬀerence between the observed data and model outputs’) were calculated as 0.87% and 91.9%, respectively, for the fuzzy logic model and the multiple regression-based model (MRM-1). To analyse the computational results in a visual manner, rela–ed graphics (including head-to-head agreement, variation of residuals and linear correlations) are also illu- strated in the estimation of pressure diﬀerence for the GT system (Figure 7). Although the ﬁrst-order polynomial model with constant term (MRM-1) produced smaller devia- tions compared to MRM-2 (the ﬁrst-order polynomial function without constant term) and MRM-3 (the exponential model) (Table 3), the conventional regression approach did not yield satisfactory predictions of the pressure diﬀerence across GT ﬁlter house as good as the proposed fuzzy logic-based model (Figure 7). 3.3. Discussion on importance and advantages of prognostic modelling in GT ﬁlter house For GT ﬁlter houses in heavy-duty power generation systems, it is very important to provide an adequate quality of combustible air without damaging the components of the entire system. On the other hand, a completely clogged GT ﬁlter house will cut oﬀ the power generation process because clogging hinders air from reaching the combustion chamber, causing a negative impact on the operation of GT systems. In such a case, a clogged ﬁlter house must be replaced as soon as possible to resume the productivity of the process. Therefore, from the engineering point of view, integration of a fuzzy logic-based prognostic model to the existing real-time system will ensure the process engineer to pre- analyse the probable eﬀects of unexpected changes that cause high-pressure diﬀerences across the ﬁlter house. Considering the nonlinear variations of both ambient conditions and GT system-related parameters, a system-adapted prognostic modelling scheme will allow the process engineer to take the necessary action in advance for the gas turbine ﬁlter houses in heavy-duty power generation systems. In this regard, developing an artiﬁcial intelligence modelling-supported fault diagnosis for GT ﬁlter houses will help to improve component life and reduce costs, resulting in an increase in the ﬁlter house lifetime. These potential economic beneﬁts will be encouraging to use artiﬁcial intelli- gence-based process control technology in other similar facilities. In the literature, many models [10,11,19,20,50–52] are proposed to predict ﬁlter pressure drop (i.e. clogging) with a quite good accuracy by only knowing the ﬂow rate, the particle size and particle mass. It is noted that the main purpose of these deterministic studies is to explore the mathematical models in prediction of the physical phenomenon such as pressure drop across the gas turbines. Additionally, there are other uncountable physical models [53–58] aimed at illustrating this mechanism, but with a wide range of MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 139 assumptions, simpliﬁcations and neglections. On the other hand, the fuzzy-based models provide a transparent and a systematic analysis for modelling highly complicated and dynamic systems [59]. This prognostic technique has the beneﬁt of the relatively straightforward sets of logical connectives rather than describing complicated mathema- tical statements for the model components in dealing with nonlinear systems [42]. It also helps to develop a sustained early-warning strategy without requiring troublesome parameter approximation procedures [36,42]. From this point of view, this kind of logical models may have a real potential to anticipate ﬁlter behaviour in complex environments or to investigate the multi-factorial phenomenon in such dynamic systems. Furthermore, it is noted that some unexpected events (e.g. ﬁre, volcanic dust, etc.) may inﬂuence the clogging of the GT ﬁlter house by changing the particle properties. The particle nature and size distribution may also strongly change with the location of the turbine (soot particles near urban areas, for example). For varying conditions (i.e. gradual and sudden changes of the model components), the importance of fuzzy logic-based approach becomes more important than conventional or deterministic models, since this method allows a rapid and straightforward prototyping in the FIS (Fuzzy Inference TM System) Editor GUI (Graphical User Interface) by using the Fuzzy Logic Toolbox within the framework of MATLAB®. This computational strategy has the ability to capture complicated inter-relationships and adapting to unforeseen changes in a dynamic and multi-component environment [42,47,60]. On the contrary of classical systems, implementation of the fuzzy logic methodology with an appropriate interactive solution algorithm also provides an opportunity to allow an adequate mapping of real multi-criteria problems quite easily in a cost-eﬀective manner [61]. 4. Conclusion A MISO fuzzy logic-based model has been proposed to make reliable estimations of pressure diﬀerence across ﬁlters used in modern, heavy-duty, gas turbine power genera- tion systems. According to the descriptive statistical analysis, the proposed prognostic methodology demonstrated more precise and eﬀective forecasts with a satisfactory determination coeﬃcient of 0.974, compared to the conventional multiple regression- based approximation. It has been clearly conﬁrmed that implementation of a fuzzy rule– based expert system provided a simple, eﬃcient and fast method in modelling of a highly nonlinear process, such as clogging of a heavy-duty GT ﬁlter house, through a set of logical rules without mathematical formulations. Actually, this is the main advantage of the proposed fuzzy-based model that it is still able to work in a straightforward manner without inquiring the high amount of trainable data. Moreover, the ability to deal with uncertainty and nonlinearity as well as the simplicity of performing a numerical char- acterization for linguistic variables makes the proposed prognostic procedure a unique and novel method compared to others for modelling of the present dynamic system. Finally, considering the usefulness of an artiﬁcial intelligence–based modelling scheme, a MIMO (multiple inputs and single output) fuzzy logic-based model (introduc- tion of additional model components and speciﬁcation of new membership functions with diﬀerent levels) will be useful to improve the proposed strategy on the GT ﬁlter houses. It is also needed to provide additional experimental data from the literature for the validity of the implemented deep learning strategy. Since such points and aspects are 140 S. A. ABDUL-WAHAB ET AL. beyond the scope of the present study, future investigations may potentially provide new insights and viewpoints into the dynamics of the GT ﬁlter house. Disclosure statement No potential conﬂict of interest was reported by the authors. ORCID Sabah Ahmed Abdul-Wahab http://orcid.org/0000-0002-5250-6340 Abubaker Sayed Mohamed Omer http://orcid.org/0000-0001-9240-2856 Kaan Yetilmezsoy http://orcid.org/0000-0003-1478-9957 Majid Bahramian http://orcid.org/0000-0002-7571-5567 References [1] G.E. Power, How a gas turbine works, 2018, https://www.gepower.com/resources/knowl edge-base/what-is-a-gas-turbine, Accessed on January 18, 2018 [2] L.S. Langston, Eﬃciency by the numbers, Mechanical Engineering Magazine, 2012, https:// memagazineblog.org/2012/07/01/eﬃciency-by-the-numbers, Accessed on January 18, 2018. [3] V.K. Mehta and R. Mehta, Principles of Power Systems, S. Chand, New Delhi, India, 2011. [4] B.B.M. 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Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis
Published: Mar 3, 2020
Keywords: Clogging phenomenon; fuzzy logic model; heavy-duty power generation system
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