Elektriksel boşalmanın bulanık mantık ile benzetimi
The simulation of electrical discharges via fuzzy logic
Yazar: KEMAL ARIKAN
Danışman: PROF. DR. ÖZCAN KALENDERLİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Elektrik Mühendisliği Ana Bilim Dalı / Elektrik Mühendisliği Bilim Dalı
Derece: Yüksek Lisans, 2013
Konu: Elektrik ve Elektronik Mühendisliği = Electrical and Electronics Engineering

Özet:
Elektriksel boşalmalar geçmişte ve günümüzde bilim insanlarının merak ettiği bir fiziksel süreç olmuş üzerinde bir çok araştırma ve kuramsal inceleme yapılmıştır. Hatta elektriksel boşalmaların yıldırıma çok benzerlik gösteren bir fiziksel olay olması sebebiyle Isaac Newton, elektriksel boşalmaları gözlemlerken, "Kıvılcımları gözlemem beni çok küçük ölçekte bir yıldırımı gözlediğimi düşünmeme neden oldu" demiştir.
Elektriksel boşalmalar yalıtkan cinsine bağlı olarak gazlarda, sıvılarda ve katılarda boşalma olmak üzere üçe ayrılır. Gazlarda düzgün alanda elektriksel boşalmayı ilk kez Townsend incelemiştir. İki düzlem elektrot arasındaki akım gerilim grafiğini gözlemleyen Townsend akımın belirli bir gerilim seviyesinden sonra üstel olarak arttığını görmüş ve elektron çığını keşfetmiştir. Townsend yoğunlaştığı araştırmaları sırasında boşalmanın kendi kendini besleme koşuluna karşı düşen delinme gerilimi bağıntısını keşfetmiştir. Ayrıca gazlarda delinme geriliminin elektrot açıklığı ve basıncın bir fonksiyonu olduğunu ilk kez Paschen keşfetmiştir. Paschen farklı gazlar için farklı basınç ve elektrot açıklığı çarpım durumlarına göre eğriler oluşturmuştur. Paschen eğrilerinin bir minimum noktası mevcuttur.
Bu tez kapsamında düzgün, az düzgün ve düzgün olmayan alanlı elektrot sistemleri için havanın farklı basınç değerleri, farklı elektrot açıklıkları ve farklı gerilim artış hızlarında delinme gerilimleri sayısal olarak hesaplanmıştır. Hesaplanan delinme gerilimi değerleri bulanık mantık yaklaşımı ile birlikte modelleme yapmak için kullanılmıştır.
Üç farklı elektrot sistemi düşünülmüştür. Bunlar karşılıklı küre-küre elektrot sistemi, düzlem-düzlem elektrot sistemi ve küre-düzlem elektrot sistemidir. Elektrotlar arası yalıtkan malzeme olarak hava düşünülmüş ve havanın basıncı için üç farklı basınç değeri olarak 760 mmHg, 660 mmHg ve 560 mmHg değerleri göz önüne alınmıştır. Elektrot açıklık değerleri ise 2 mm, 4 mm ve 6 mm olarak değiştirilmiştir. Gerilim artış hızı olarak da Townsend'in kendi kendini besleme denkleminin paydasında bulunan ikinci iyonizasyon katsayısı için üç farklı değer alınmıştır. Üç farklı elektrot sistemi, üç farklı gaz basıncı, üç farklı elektrot açıklığı ve üç farklı iyonizasyon katsayısı ile seksenbir farklı durum için seksenbir farklı delinme gerilimi etkin değer cinsinden hesapla bulunmuştur.
Elektrot sistemlerini birbirinden ayırt etmek için gereken alan düzgünlük faktörü değerleri, sonlu elemanlar yöntemini esas alan FEMM adlı bilgisayar programı yardımıyla hesaplanmıştır. Sayısal olarak hesaplanmayan ara değerler ise birinci dereceden interpolasyon ile programlanarak hesaplanmıştır.
Tezde altı farklı bulanık çıkarım sistemi tasarlanmış olup, üç tanesi delinme geriliminin tespiti için, diğer üç tanesi elektrot açıklığına ve elektrot sistemine bağlı olarak alan düzgünlük faktörlerinin değişimini göstermektedir.
Tasarlanan bulanık çıkarım sistemleri, tek bir model üzerinden kullanılabilmesi amacıyla Matlab/Simulink modeline entegre edilmiştir. 
Sayısal olarak elde edilen delinme gerilim değerleri ile bulanık mantık değerlerinden elde edilen delinme gerilimleri arasında hata oranı hesaplanmıştır.

Abstract:
Electrical discharge phenomena have been an important physical process from past to present so the scientists have done the many researches and theoretical examination upon it. Moreover, because of the similarity between lightning and electrical discharges, Isaac Newton who is important physicist when he was observing electrical discharges he said that observing sparks caused to think me about small scale lightning.
If a voltage which is applied on an insulator exceeds a certain threshold voltage value, the whole part of the insulator or specific areas becomes conductive path; this process is called electrical discharge. Electrical discharges divided into three depending on the type of insulating material these are discharge in gases, liquids and solids. Electrical discharges in gases in the uniform field configuration have examined by Townsend for the first time. Townsend observed the change of the current and voltage levels in a parallel two plane electrodes. He discovered an exponential increase in the current graph beyond a certain voltage level, so he thought that the reason of this non-linearity is electron avalanche process in the discharge phenomenon. According to the ionisation process in gases, an electron releases from cathode follows the path to anode. Each electron releases from the cathode have possibility to ionise the neutral atoms or molecules. The atoms or molecules being ionised may releases different electrons and this process seems like an avalanche. Townsend found a coefficient called Townsend's first coefficient (?). On the other hand, if applied source between two points is removed, discharge cannot be self-sustaining. For this reason, Townsend thought that an electron releases from cathode must creates a reverse until it reaches to anode. Townsend second coefficient ? defines number of electrons releases from cathode by a positive ion. By this way, discharge will be self-sustaining the surveys that Townsend concentrated on caused discovering the self-sustaining breakdown voltage equation for uniform electrode systems.
On the other hand, Paschen the first time discovered the production of the electrode gap and gas pressure is a function of the breakdown voltage. Paschen did lots of experiments on different gases. He discovered a curve called Paschen curve, has a minimum. The curve parameters are breakdown voltage and production of the pressure and electrode gap spacing.
In this thesis, for the uniform, weakly non-uniform and non-uniform electrode systems, as an insulating material air breakdown voltage for different pressure levels, different electrode spacing gaps and different voltage rise rates calculated analytically. The analytically calculated breakdown voltages were used modeling using fuzzy logic approach.
In recent years especially in Japan, America and Germany approximately more than one thousands fuzzy logic application performed successfully. Washing machines, microwave ovens, cameras, refrigerators, credit cards are some of the applications in consumer products. Other applications of fuzzy logic are industrial process control, medical instrumentation, decision support systems and portfolio selection.
In an industrial process control, structure and the dynamic properties of the systems should be known very well and their mathematical models are need to known. But, the parameters of the systems sometimes can not to be understood easily and be unknown. In such cases usually the problem is solved by an expert person. The expert uses daily expressions using by human beings such as little, very little, too much. These linguistic expressions can be translated to machine language. Thus, fuzzy logic is founded on such relations. For fuzzy logic, it can be said adapting math to real world. The term "fuzzy logic" first introduced by Lotfi A. Zadeh in 1965 with his proposal of fuzzy set theory.
The general properties of fuzzy logic are explained by Zadeh as follows. In fuzzy logic approximate thinking is used instead of thinking based on precise value. In fuzzy logic everything can be shown with a precise degree in interval [0, 1] In fuzzy logic, knowledge is using linguistic expressions such as big, small, very little. Fuzzy inference process is done by rules defined between linguistic expressions. Every logical system can be expressed as fuzzy. Fuzzy logic is very useful for systems which their mathematical modeling is very difficult.
The first application of fuzzy logic is applied by Mamdani in 1974. Mamdani controlled a steam engine by fuzzy logic.
Mamdani's method is the most commonly used fuzzy inference process and it's operation is as follows. Firstly the raw input data from the external world may be required pre-processing such as normalizing or scaling. After that, fuzzification of the inputs is next step. It is essential to associate each input with a fuzzy set group. These fuzzy sets should span the entire range of values associated with its input. System designer determines how many fuzzy sets per input. The degree of membership in each of an input's fuzzy set is defined by the membership functions. These functions are also designed by system designer. There are some simple shapes such as triangle, trapezoids, Gaussian curves to define the membership functions.
To produce the fuzzy outputs (called consequent sets) fuzzy rules combines the fuzzy inputs. Description of the action of the fuzzy systems can be decided by these rules. Each rule has the form of IF/THEN statement and the input sets can be combined by logical operators such as AND (fuzzy intersection), OR (fuzzy union) and NOT (fuzzy complement). The result for each rule has a single number changes between 0 and 1. This indicates the degree of support for the rule. Implication process is the following process. After that by aggregation process all the outputs of each rule unified into a single fuzzy set with three methods: maximum, probabilistic OR and union. Finally, the fuzzy set which includes a range of output values is converted into a single number which can be useful for the external world. The most common defuzzification method is centre of gravity calculation.
In the model, three different electrode configurations were considered these are parallel plane electrodes, sphere-sphere electrodes, and sphere-plane electrodes. The electrodes are considered to be positioned in a cylindrical container its height 50 cm and bottom diameter 10 cm. Plane electrodes are solid model with its bottom diameter 5 cm, top diameter 3.5 cm and 0.75 cm height. Diameters of the sphere electrodes are 5 cm. As an insulating material air was used, and air pressure changes from 760 mmHg to 560 mmHg decreasing 100 mmHg steps. Three different electrode gap spacings were considered. These are 2 mm, 4 mm and 6 mms. The self-sustaining breakdown voltage equation for uniform electrode systems (Townsend breakdown equation) can only be applied for plane-plane electrode system. So it is essential to transform the equation for quasi uniform (sphere-plane) and non-uniform (sphere-sphere) electrode configurations. For this reason, the electrode gap spacing being in the equation can be rewritten. It is useful to know some analytical expressions like field utilisation factor (?), maximum electric field (Emax), mean electric field (Emean), equivalent gap spacing (?) for each electrode configuration. The mean electric field for all electrode configurations is the ratio of voltage applied between electrodes to electrode gap spacing which is the nearest distance between electrodes. Emean = V/a. Maximum electric field for non-uniform electrode configurations is Emax = V/?. The electrode utilization factor is the ratio of mean electric field to maximum electric field, ? = Emean / Emax = (V/a)/(V/?) = ?/a. The uniformity factors of electrode systems were calculated by a computer programme called FEMM based on finite element method. Equivalent gap spacing for non-uniform electrode configurations are calculated analytically. Thus the replacement of the electrode gap spacing with ?/? provides to calculate analytical breakdown voltages for sphere-sphere and sphere-plane electrode configurations. The voltage rise rate has three different combinations. It changes 1 kV/s, 2 kV/s, 3 kV/s. For this purpose Townsend second coefficient (?) in the denominator of the Townsend equation took three different values. Normally ? can take value between 1/50 and 1/5000. ? refers to number of electrons released from cathode by a positive ion. The breakdown voltage of same electrode configuration only voltage rise rates are different, takes different values, for example for the same configuration when voltage rise rate is slowest (1 kV/s) it takes the minimum value. When voltage rise rate is fastest (3 kV/s), value of the breakdown voltage is maximum. From this explanation, in the thesis when voltage rise rate is 1 kV/s, ? took maximum value 1/50; when voltage rise rate is 2 kV/s, ? took 1/500; when voltage rise rate is 3 kV/s, ? took 1/5000. Finally, from eighty-one different combinations, eighty-one different breakdown voltages were calculated analytically as effective value. 
In the thesis, six different fuzzy inference systems were designed. Three of them were used to find breakdown values, others were used to find the uniformity of electrode systems depending of the electrode configuration and electrode gap spacing. In order to find breakdown voltage, input parameters in fuzzy inference systems are chosen product of pressure and gap spacing, and voltage rise rate. On the other hand input parameters are chosen electrode gap spacing to find electrode utilization factors in other three fuzzy inference systems. In order to estimate breakdown voltages with Townsend breakdown equation as data for input parameters, these estimated values corresponding to product of pressure and electrode gap spacing are interpolated with first order interpolation using programming with MATLAB for linearity.
Designed fuzzy inference systems were integrated in Matlab/Simulink model to be used on a single model for comparison between different electrode configurations with each other. In the conclusion part of the thesis, the errors between the breakdown voltage values obtained from the analytical calculation and fuzzy logic approach were calculated with obtained Simulink model and listed. The error values are not exceeded 3%, so fuzzy logic modeling is resulted successfully. Moreover, how the errors can be reduced is mentioned in the conclusion section.